Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 864865 11 pageshttpdxdoiorg1011552013864865
Research ArticleA Fourth-Order Block-Grid Method forSolving Laplacersquos Equation on a Staircase Polygonwith Boundary Functions in 119862119896120582
A A Dosiyev and S Cival Buranay
Department of Mathematics Eastern Mediterranean University Gazimagusa North Cyprus Mersin 10 Turkey
Correspondence should be addressed to A A Dosiyev adiguzeldosiyevemuedutr
Received 30 April 2013 Accepted 11 May 2013
Academic Editor Allaberen Ashyralyev
Copyright copy 2013 A A Dosiyev and S Cival Buranay This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
The integral representations of the solution around the vertices of the interior reentered angles (on the ldquosingularrdquo parts) areapproximated by the composite midpoint rule when the boundary functions are from 119862
4120582 0 lt 120582 lt 1 These approximations
are connected with the 9-point approximation of Laplacersquos equation on each rectangular grid on the ldquononsingularrdquo part of thepolygon by the fourth-order gluing operator It is proved that the uniform error is of order 119874(ℎ
4+ 120576) where 120576 gt 0 and ℎ is the
mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between the approximate and the exact solutions in each ldquosingularrdquo part 119874((ℎ
4+ 120576)119903
1120572119895minus119901
119895) order is obtained here 119903
119895is the distance from the current point to the vertex in question and 120572
119895120587
is the value of the interior angle of the 119895th vertex Numerical results are given in the last section to support the theoretical results
1 Introduction
In the last two decades among different approaches to solvethe elliptic boundary value problems with singularities aspecial emphasis has been placed on the construction ofcombined methods in which differential properties of thesolution in different parts of the domain are used (see [1 2]and references therein)
In [2ndash7] a new combined difference-analytical methodcalled the block-grid method (BGM) is proposed for thesolution of the Laplace equation on polygons when theboundary functions on the sides causing the singular verticesare given as algebraic polynomials of the arc length In theBGM the given polygon is covered by a finite number ofoverlapping sectors around the singular vertices (ldquosingularrdquoparts) and rectangles for the part of the polygon which lies ata positive distance from these vertices (ldquononsingularrdquo part)The special integral representation in each ldquosingularrdquo partis approximated and they are connected by the appropriateorder gluing operator with the finite difference equationsused in the ldquononsingularrdquo part of the polygon
In [8 9] the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices in the BGM was removed It was assumedthat the boundary function on each side of the polygon isgiven from the Holder classes 119862119896120582
0 lt 120582 lt 1 and on theldquononsingularrdquo part the 5-point scheme is used when 119896 = 2
[8] and the 9-point scheme is used when 119896 = 6 [9] For the5-point scheme a simple linear interpolation with 4 pointsis used For the 9-point scheme an interpolation with 31points is used to construct a gluing operator connectingthe subsystems Moreover to connect the quadrature nodeswhich are at a distance of less than 4ℎ from boundary of thepolygon a special representation of the harmonic functionthrough the integrals of Poisson type for a half plane is used(see [9])
In this paper the BGM is developed for the Dirichletproblem when the boundary function on each side of thepolygon is from 119862
4120582 by using the 9-point scheme on theldquononsingularrdquo part with 16-point gluing operator for allquadrature nodes including those near the boundary The
2 Abstract and Applied Analysis
paper is organized as follows in Section 2 the boundaryvalue problem and the integral representations of the exactsolution in each ldquosingularrdquo part are given In Section 3 tosupport the aim of the paper a Dirichlet problem on therectangle for the known exact solution from 119862
119896120582 119896 = 3 4 issolved using the 9-point scheme and the numerical results areillustrated In Section 4 the system of block-grid equationsand the convergence theorems are given In Section 5 a highlyaccurate approximation of the coefficient of the leadingsingular term of the exact solution (stress intensity factor) isgiven In Section 6 the method is illustrated for solving theproblem in L-shaped polygonwith the corner singularityTheconclusions are summarized in Section 7
2 Dirichlet Problem on a Staircase Polygon
Let 119866 be an open simply connected polygon 120574119895 119895 =
1 2 119873 its sides including the ends enumerated coun-terclockwise 120574 = 120574
1cup sdot sdot sdot cup 120574
119873the boundary of 119866 and 120572
119895120587
(120572119895= 12 1 32 2) the interior angle formed by the sides
120574119895minus1
and 120574119895 (120574
0= 120574
119873) Denote by 119860
119895= 120574
119895minus1cap 120574
119895the vertex of
the 119895th angle and by 119903119895 120579
119895a polar system of coordinates with a
pole in119860119895 where the angle 120579
119895is taken counterclockwise from
the side 120574119895
We consider the boundary value problem
Δ119906 = 0 on 119866 119906 = 120593119895(119904) on 120574
119895 1 le 119895 le 119873 (1)
whereΔ equiv 1205972120597119909
2+120597
2120597119910
2 120593
119895is a given function on 120574
119895of the
arc length 119904 taken along 120574 and 120593119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 that
is 120593119895has the fourth-order derivative on 120574
119895 which satisfies a
Holder condition with exponent 120582At some vertices119860
119895 (119904 = 119904
119895) for 120572
119895= 12 the conjugation
conditions
120593(2119902)
119895minus1(119904
119895) = (minus1)
119902120593(2119902)
119895(119904
119895) 119902 = 0 1 (2)
are fulfilled For the remaining vertices 119860119895 the values of 120593
119895minus1
and 120593119895at 119860
119895might be different Let 119864 be the set of all 119895 (1 le
119895 le 119873) for which 120572119895
= 12 or 120572119895= 12 but (2) is not fulfilled
In the neighborhood of 119860119895 119895 isin 119864 we construct two fixed
block sectors 119879119894
119895= 119879
119895(119903
119895119894) sub 119866 119894 = 1 2 where 0 lt 119903
1198952lt 119903
1198951lt
min119904119895+1
minus119904119895 119904
119895minus119904
119895minus1 119879
119895(119903) = (119903
119895 120579
119895) 0 lt 119903
119895lt 119903 0 lt 120579
119895lt
120572119895120587Let (see [10])
1205931198950(119905) = 120593
119895(119904
119895+ 119905) minus 120593
119895(119904
119895)
1205931198951(119905) = 120593
119895minus1(119904
119895minus 119905) minus 120593
119895minus1(119904
119895)
(3)
119876119895(119903
119895 120579
119895) = 120593
119895(119904
119895) +
(120593119895minus1
(119904119895) minus 120593
119895(119904
119895)) 120579
119895
120572119895120587
+1
120587
1
sum
119896=0
int
120590119895119896
0
119910119895120593119895119896
(119905120572119895) 119889119905
(119905 minus (minus1)119896119909119895)2
+ 1199102
119895
(4)
where
119909119895= 119903
1120572119895
119895cos(
120579119895
120572119895
) 119910119895= 119903
1120572119895
119895sin(
120579119895
120572119895
)
120590119895119896
=10038161003816100381610038161003816119904119895+1minus119896
minus 119904119895minus119896
10038161003816100381610038161003816
1120572119895
(5)
The function119876119895(119903
119895 120579
119895) is harmonic on119879
1
119895and satisfies the
boundary conditions in (1) on 120574119895minus1
cap 1198791
119895and 120574
119895cap 119879
1
119895 119895 isin 119864
except for the point 119860119895when 120593
119895minus1(119904
119895) = 120593
119895(119904
119895)
We formally set the value of 119876119895(119903
119895 120579
119895) and the solution
119906 of the problem (1) at the vertex 119860119895equal to (120593
119895minus1(119904
119895) +
120593119895(119904
119895))2 119895 isin 119864Let
119877119895(119903 120579 120578)
=1
120572119895
1
sum
119896=0
(minus1)119896119877((
119903
1199031198952
)
1120572119895
120579
120572119895
(minus1)119896 120578
120572119895
)
119895 isin 119864
(6)
where
119877 (119903 120579 120578) =1 minus 119903
2
2120587 (1 minus 2119903 cos (120579 minus 120578) + 1199032)(7)
is the kernel of the Poisson integral for a unit circle
Lemma 1 (see [10]) The solution 119906 of the boundary valueproblem (1) can be represented on 119879
2
119895 119881
119895 119895 isin 119864 in the form
119906 (119903119895 120579
119895) = 119876
119895(119903
119895 120579
119895)
+ int
120572119895120587
0
119877119895(119903
119895 120579
119895 120578) (119906 (119903
1198952 120578) minus 119876
119895(119903
1198952 120578)) 119889120578
(8)
where 119881119895is the curvilinear part of the boundary of 1198792
119895 and
119876119895(119903
119895 120579
119895) is the function defined by (4)
3 9-Point Solution on Rectangles
Let Π = (119909 119910) 0 lt 119909 lt 119886 0 lt 119910 lt 119887 be a rectanglewith 119886119887 being rational 120574
119895 119895 = 1 2 3 4 the sides including
the ends enumerated counterclockwise starting from the leftside (120574
0equiv 120574
4 120574
5equiv 120574
1) and 120574 = cup
4
119895=1120574119895the boundary of Π
We consider the boundary value problem
Δ119906 = 0 on Π
119906 = 120593119895
on 120574119895 119895 = 1 2 3 4
(9)
where 120593119895is the given function on 120574
119895
Definition 2 One says that the solution 119906 of the problem (9)belongs to 119862
4120582(Π) if
120593119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 119895 = 1 2 3 4 (10)
Abstract and Applied Analysis 3
and at the vertices 119860119895= 120574
119895minus1cap 120574
119895the conjugation conditions
120593(2119902)
119895= (minus1)
119902120593(2119902)
119895minus1 119902 = 0 1 (11)
are satisfied
Remark 3 From Theorem 31 in [11] it follows that the classof functions 1198624120582
(Π) is wider than 1198624120582
(Π)
Let ℎ gt 0 with 119886ℎ ge 2 119887ℎ ge 2 integers We assignto Π
ℎ a square net on Π with step ℎ obtained with the lines119910 = 0 ℎ 2ℎ Let 120574ℎ
119895be a set of nodes on the interior of 120574
119895
and let
120574ℎ
119895= 120574
119895cap 120574
119895+1 120574
ℎ= cup
4
119895=1(120574
ℎ
119895cup 120574
ℎ
119895)
Πℎ
= Πℎcup 120574
ℎ
(12)
We consider the system of finite difference equations
119906ℎ= 119861119906
ℎon Π
ℎ
119906ℎ= 120593
119895on 120574
ℎ
119895 119895 = 1 2 3 4
(13)
where
119861119906 (119909 119910)
equiv(119906 (119909 + ℎ 119910) + 119906 (119909 119910 + ℎ) + 119906 (119909 minus ℎ 119910) + 119906 (119909 119910 minus ℎ))
5
+ ( ((119906 (119909 + ℎ 119910 + ℎ) + 119906 (119909 minus ℎ 119910 + ℎ)
+ 119906 (119909 minus ℎ 119910 minus ℎ) + 119906 (119909 + ℎ 119910 minus ℎ)))
times 20minus1)
(14)
On the basis of the maximum principle the uniquesolvability of the system of finite difference equations (13)follows (see [12 Chapter 4])
Everywhere below we will denote constants which areindependent of ℎ and of the cofactors on their right by119888 119888
0 119888
1 generally using the same notation for different
constants for simplicity
Theorem 4 Let 119906 be the solution of problem (9) If 119906 isin
1198624120582
(Π) then
maxΠℎ
1003816100381610038161003816119906ℎ minus 1199061003816100381610038161003816 le ch4 (15)
where 119906ℎis the solution of the system (13)
Proof For the proof of this theorem see [13]
LetΠ1015840= (119909 119910) minus025 lt 119909 lt 025 0 lt 119910 lt 1 and let 1205741015840
be the boundary of Π1015840 We consider the Dirichlet problem
Δ119906 = 0 on Π1015840
119906 = V on 1205741015840
(16)
0 20 40 60 80 100 120 14011
12
13
14
15
16
17
119862400003
11986240003
1198624003
1198623055
1198623066
1198623075
119862308
Rϱ
hminus1 = 2ϱ
Figure 1 Dependence of the approximate solutions for the bound-ary functions from 119862
119896120582
where V = 119903119896+120582 cos(119896 + 120582)120579 119903 = radic1199092 + 1199102 0 lt 120582 lt 1 is the
exact solution of this problem which is from 119862119896120582
(Π1015840
)We solve the problem (16) by approximating 9-point
scheme when 119896 = 3 4 for the different values of 120582In Figure 1 the order of numerical convergence
R984858
Πℎ =
maxΠℎ10038161003816100381610038161199062minus984858 minus 119906
1003816100381610038161003816
maxΠℎ10038161003816100381610038161199062minus(984858+1) minus 119906
1003816100381610038161003816
(17)
of the 9-point solution 119906ℎ for different ℎ = 2
minus984858 and 984858 =
4 5 6 7 is demonstratedThese results show that the order ofnumerical convergence when the exact solution 119906 isin 119862
119896120582(Π)
depends on 119896 and 120582 and is119874(ℎ4)when 119896 = 4 which supports
estimation (15) Moreover this dependence also requires theuse of fourth-order gluing operator for all quadrature nodesin the construction of the system of block-grid equationswhen the given boundary functions are from the Holderclasses 1198624120582
4 System of Block-Grid Equations
In addition to the sectors 1198791
119895and 119879
2
119895(see Section 2) in the
neighborhood of each vertex 119860119895 119895 isin 119864 of the polygon 119866 we
construct two more sectors 1198793
119895and 119879
4
119895 where 0 lt 119903
1198954lt 119903
1198953lt
1199031198952 119903
1198953= (119903
1198952+ 119903
1198954)2 and 119879
3
119896cap 119879
3
119897= 0 119896 = 119897 119896 119897 isin 119864 and let
119866119879= 119866 (cup
119895isin1198641198794
119895)
We cover the given solution domain (a staircase polygon)by the finite number of sectors 119879
1
119895 119895 isin 119864 and rectangles
Π119896
sub 119866119879 119896 = 1 2 119872 as is shown in Figure 2 for
the case of 119871-shaped polygon where 119895 = 1119872 = 4 (seealso [2]) It is assumed that for the sides 119886
1119896and 119886
2119896of
Π119896the quotient 119886
1119896119886
2119896is rational and 119866 = (cup
119872
119896=1Π
119896) cup
(cup119895isin119864
1198793
119895) Let 120578
119896be the boundary of the rectangleΠ
119896 let119881
119895be
4 Abstract and Applied Analysis
minus1 minus05 0 05 1minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
Π1
Π212057810
12057820
119886
119887119888
119889119890
120579
1198601 1205741
1205740
11987941
11987931
11987921
11987911
Π3
Π4
12057820
12057830
12057830
12057840
119903
11990311
11990312
11990313 11990314
119866119878
Figure 2 Description of BGM for the L-shaped domain
the curvilinear part of the boundary of the sector 1198792
119895 and let
119905119896119895
= 120578119896cap 119879
3
119895 We choose a natural number 119899 and define
the quantities 119899(119895) = max4 [120572119895119899] 120573
119895= 120572
119895120587119899(119895) and
120579119898
119895= (119898 minus 12)120573
119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881
119895
we take the points (1199031198952 120579
119898
119895) 1 le 119898 le 119899(119895) and denote the
set of these points by 119881119899
119895 We introduce the parameter ℎ isin
(0 12060004] where 120600
0is a gluing depth of the rectanglesΠ
119896 119896 =
1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872
withmaximal possible step ℎ119896le minℎmin119886
1119896 119886
21198966 such
that the boundary 120578119896lies entirely on the grid lines Let Πℎ
119896be
the set of grid nodes on Π119896 let 120578ℎ
119896be the set of nodes on 120578
119896
and let Πℎ
119896= Π
ℎ
119896cup 120578
ℎ
119896 We denote the set of nodes on the
closure of 120578119896cap 119866
119879by 120578
ℎ
1198960 the set of nodes on 119905
119896119895by 119905
ℎ
119896119895 and
the set of remaining nodes on 120578119896by 120578
ℎ
1198961
Let
120596ℎ119899
= (cup119872
119896=1120578ℎ
1198960) cup (cup
119895isin119864119881
119899
119895)
119866ℎ119899
119879= 120596
ℎ119899cup (cup
119872
119896=1Π
ℎ
119896)
(18)
Let 120593 = 120593119895119873
119895=1 where 120593
119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 is
the given function in (1) We use the matching operator 1198784
at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906
ℎ 120593) at the point 119875 isin 120596
ℎ119899 is expressed linearly in termsof the values of 119906
ℎat the points 119875
119896of the grid constructed on
Π119896(119875)
(119875 isin Π119896(119875)
) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)
119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906
ℎ 0) has the representation
1198784(119906
ℎ 0) = sum
0le119897le15
120585119897119906ℎ119897 (19)
where 119906ℎ119896
= 119906ℎ(119875
119896)
120585119897ge 0 sum
0le119897le15
120585119897= 1 (20)
119906 minus 1198784(119906 120593) = 119874 (ℎ
4) (21)
Let 120596ℎ119899
119868sub 120596
ℎ119899 be the set of such points 119875 isin 120596ℎ119899
for which all points 119875119897in expression (19) are in cup
119872
119896=1Π
ℎ
119896 If
some points 119875119897in (19) emerge through the side 120574
119898 then the
set of such points 119875 is denoted by 120596ℎ119899
119863 According to the
construction of 1198784 in [14] the expression 1198784(119906
ℎ 120593) at each
point 119875 isin 120596ℎ119899
= 120596ℎ119899
119868cup 120596
ℎ119899
119863can be expressed as follows
1198784(119906
ℎ 120593)
=
1198784119906ℎ 119875 isin 120596
ℎ119899
119868
1198784(119906
ℎminus
3
sum
119896=0
119886119896Re 119911119896) + (
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(22)
where
119886119896=
1
119896
119889119896120593119898(119904)
119889119904119896
100381610038161003816100381610038161003816100381610038161003816119904=119904119875
119896 = 0 1 2 3 (23)
and 119904119875corresponds to such a point119876 isin 120574
119898for which the line
119875119876 is perpendicular to 120574119898
Let
119876119895= 119876
119895(119903
119895 120579
119895) 119876
119902
1198952= 119876
119895(119903
1198952 120579
119902
119895) (24)
The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876
120576
119895and 119876
119902120576
1198952of the
quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876
120576
119895minus 119876
119895
10038161003816100381610038161003816le 119888
1120576
10038161003816100381610038161003816119876
119902120576
1198952minus 119876
119902
1198952
10038161003816100381610038161003816le 119888
2120576 (25)
where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888
2are constants
independent of 120576Consider the system of linear algebraic equations
119906120576
ℎ= 119861119906
120576
ℎon Π
ℎ
119896
119906120576
ℎ= 120593
119898on 120578
ℎ
1198961cap 120574
119898
119906120576
ℎ(119903
119895 120579
119895)
= 119876120576
119895+ 120573
119895
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
times 119877119895(119903
119895 120579
119895 120579
119902
119895) on (119903
119895 120579
119895) isin 119905
ℎ
119896119895
119906120576
ℎ= 119878
4119906120576
ℎon 120596
ℎ119899
(26)
where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864
Abstract and Applied Analysis 5
Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2
minus984858 and 120576 = 5 times 10minus13
(2minus984858 119899) 120577
120576
ℎ119866119873119878 R
984858
119866119873119878
(2minus4 60) 1609 times 10
minus815577
(2minus5 170) 1033 times 10
minus9
(2minus5 130) 1191 times 10
minus916690
(2minus6 150) 7136 times 10
minus11
(2minus5 140) 1136 times 10
minus916259
(2minus6 170) 6991 times 10
minus11
(2minus6 100) 2169 times 10
minus1017096
(2minus7 130) 1269 times 10
minus11
Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =
2minus984858 and 120576 = 5 times 10
minus13
(2minus984858 119899) 120577
120576
ℎ119866119878 R
984858
119866119878
(2minus4 100) 1931 times 10
minus816078
(2minus5 150) 1182 times 10
minus9
(2minus5 130) 1294 times 10
minus916789
(2minus6 150) 7708 times 10
minus11
(2minus5 140) 1312 times 10
minus917967
(2minus6 170) 7304 times 10
minus11
(2minus6 100) 2389 times 10
minus1018164
(2minus7 130) 1315 times 10
minus11
Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) 120577
120576
ℎ119866119873119878 120577
120576
ℎ119866119878 Iteration
(16 70) 1139 times 10minus8
1572 times 10minus8
22
(32 170) 1033 times 10minus9
1184 times 10minus9
23
(64 170) 6990 times 10minus11
7304 times 10minus11
24
(128 200) 8628 times 10minus12
8833 times 10minus12
25
Let 119879lowast
119895= 119879
119895(119903
lowast
119895) be the sector where 119903lowast
119895= (119903
1198952+119903
1198953)2 119895 isin
119864 and let 119906120576ℎ(119903
1198952 120579
119902
119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution
values of the system (26) on 119881ℎ
119895(at the quadrature nodes)
The function
119880120576
ℎ(119903
119895 120579
119895) = 119876
119895(119903
119895 120579
119895)
+ 120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) (119906
120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
(27)
defined on 119879lowast
119895 is called an approximate solution of the
problem (1) on the closed block 1198793
119895 119895 isin 119864
Definition 5 The system (26) and (27) is called the system ofblock-grid equations
Theorem 6 There is a natural number 1198990 such that for all
119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution
Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899
0and 120590 such that for all 119899 ge 119899
0
max(119903119895120579119895)isin119879
3
119895
120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) le 120590 lt 1 (28)
The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)
Theorem 7 There exists a natural number 1198990 such that for all
119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)
where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid
max119866ℎ119899
119879
1003816100381610038161003816119906120576
ℎminus 119906
1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816le 119888
119901(ℎ
4+ 120576) on 119879
3
119895
(31)
for integer 1120572119895when 119901 ge 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895
(32)
for any 1120572119895 if 0 le 119901 lt 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895 119860
119895
(33)
for noninteger 1120572119895 when 119901 gt 1120572
119895 Everywhere 0 le 119902 le 119901 119906
is the exact solution of the problem (1) and119880120576
ℎ(119903
119895 120579
119895) is defined
by formula (27)
Proof Let
120585120576
ℎ= 119906
120576
ℎminus 119906 (34)
where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866
ℎ119899
119879
of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576
ℎsatisfies the system of difference equations
120585120576
ℎ= 119861120585
120576
ℎ+ 119903
1
ℎon Π
ℎ
119896
120585120576
ℎ= 0 on 120578
ℎ
1198961
120585120576
ℎ(119903
119895 120579
119895) = 120573
119895
119899(119895)
sum
119902=1
120585120576
ℎ(119903
1198952 120579
119902
119895) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
+ 1199032
119895ℎ (119903
119895 120579
119895) isin 119905
ℎ
119896119895
120585120576
ℎ= 119878
4120585120576
ℎ+ 119903
3
ℎon 120596
ℎ119899
(35)
6 Abstract and Applied Analysis
Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 3895 times 10minus7
3895 times 10minus7
(32 170) 4627 times 10minus8
4627 times 10minus8
(64 170) 1124 times 10minus9
3125 times 10minus9
(128 200) 2214 times 10minus10
2233 times 10minus10
Table 5 In119866119878 the minimum errors of the derivatives over the pairs
(ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 9663 times 10minus6
9663 times 10minus6
(32 170) 9653 times 10minus6
9653 times 10minus6
(64 170) 9649 times 10minus6
9649 times 10minus6
(128 200) 9648 times 10minus6
9648 times 10minus6
where 1 le 119896 le 119872 119895 isin 119864
1199031
ℎ= 119861119906 minus 119906 on cup
119872
119896=1Π
ℎ
119896 (36)
1199032
119895ℎ= 120573
119895
119899(119895)
sum
119902=1
(119906 (1199031198952 120579
119902
119895) minus 119876
119902120576
1198952) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
minus (119906 minus 119876120576
119895) on cup
119872
119896=1(cup
119895isin119864119905ℎ
119896119895)
(37)
1199033
ℎ
=
1198784119906 minus 119906 on 120596
ℎ119899
119868
1198784(119906 minus
3
sum
119896=0
119886119896Re 119911119896) minus (119906 minus
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(38)
On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows
The function 119880120576
ℎ(119903
119895 120579
119895) given by formula (27) defined
on the closed sector 119879lowast
119895 119895 isin 119864 where 119903
lowast
119895= (119903
1198952+ 119903
1198953)2
and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879
2
119895 119881
119895 119895 isin 119864 and then the
difference function 120577120576
ℎ(119903
119895 120579
119895) = 119880
120576
ℎ(119903
119895 120579
119895)minus119906(119903
119895 120579
119895) is defined
on 119879lowast
119895 119895 isin 119864 and
120577120576
ℎ(119903
lowast
119895 0) = 120577
120576
ℎ(119903
lowast
119895 120572
119895120587) = 0 119895 isin 119864 (39)
On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge
max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain
10038161003816100381610038161003816120577120576
ℎ(119903
119895 120579
119895)10038161003816100381610038161003816le 119888 (ℎ
4+ 120576) on 119879
lowast
119895 119895 isin 119864 (40)
Furthermore the function 120577120576
ℎ(119903
119895 120579
119895) continuous on 119879
lowast
119895is a
solution of the following Dirichlet problem
Δ120577120576
ℎ= 0 on 119879
lowast
119895
120577120576
ℎ= 0 on 120574
119898cap 119879
lowast
119895 119898 = 119895 minus 1 119895
120577120576
ℎ(119903
lowast
119895 120579
119895) = 119880
120576
ℎ(119903
lowast
119895 120579
119895) minus 119906 (119903
lowast
119895 120579
119895) 0 le 120579
119895le 120572
119895120587
(41)
Since 1198793
119895sub 119879
lowast
119895 on the basis of (39) and (40) from Lemma
612 in [16] inequalities (31)ndash(33) of Theorem 7 follow
5 Stress Intensity Factor
Let in the condition 120593119895isin 119862
4120582(120574
119895) the exponent 120582 be such
that
120572119895(4 + 120582) = 0 2120572
119895(4 + 120582) = 0 (42)
where sdot is the symbol of fractional partThese conditions forthe given 120572
119895can be fulfilled by decreasing 120582
On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879
lowast
119895 119895 isin 119864 as follows
119906 (119909119895 119910
119895) = (119909
119895 119910
119895) +
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911
+
119899120572119895
sum
119896=1
120591(119895)
119896119903119896120572119895
119895sin
119896120579119895
120572119895
(43)
where 119899120572119895
= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909
119895+119894119910
119895 120583(119895)
119896
and 120591(119895)
119896are some numbers and (119909
119895 119910
119895) isin 119862
4120582(119879
2
119895) is the
harmonic on 1198792
119895 By taking 120579
119895= 120572
1198951205872 from the formula
(43) it follows that the coefficient 120591(119895)1
which is called the stressintensity factor can be represented as
120591(119895)
1= lim
119903119895rarr0
1
1199031120572119895
119895
(119906 (119909119895 119910
119895) minus (119909
119895 119910
119895)
minus
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911)
(44)
Abstract and Applied Analysis 7
Table 6 The stress intensity factor 1205911205981119899
for 119899 = 70 170 200 when 120576 = 5 times 10minus13
ℎminus1
120591120576
170120591120576
1170120591120576
1200
16 1000000014856688 1000000017180415 1000000017197438
32 1000000005800844 1000000001230267 1000000001236709
64 1000000004138169 1000000000073107 1000000000079938
128 1000000004053153 1000000000003531 1000000000003267
10minus15 10minus10 10minus5 10010minus10
10minus5
100
120576
Max
erro
r
ℎ = 116 119899 = 70
(a)
10minus10
10minus5
100
10minus15 10minus10 10minus5 100
120576
Max
erro
r
ℎ = 132 119899 = 170
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 3 Dependence on 120576 for ℎminus1 = 16 32
From formula (44) it follows that 120591(119895)
1can be approxi-
mated by
120591(119895)120576
1119899
= lim119903119895rarr0
1
1199031120572119895
119895
(119880120576
ℎ(119903
119895 120579
119895)
minus(120593119895(119904
119895) + (120593
119895minus1(119904
119895) minus 120593
119895(119904
119895))
120579119895
120572119895120587))
(45)
Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula
120591(119895)120576
1119899=
1
120587int
1205901198950
0
1205931198950(119905
120572119895) 119889119905
1199052+
1
120587int
1205901198951
0
1205931198951(119905
120572119895) 119889119905
1199052
+2
119899 (119895) 1199031120572119895
1198952
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952) sin 1
120572119895
120579119902
119895
(46)
This formula is obtained for the second-order BGM in [8]
10minus15 10minus10 10minus5 100
120576
h = 164 n = 170100
10minus10
10minus20
Max
erro
r
(a)
10minus15 10minus10 10minus5 100
120576
h = 1128 n = 200100
10minus10
10minus20
Max
erro
r
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 4 Dependence on 120576 for ℎminus1 = 64 128
6 Numerical Results
Let 119866 be L-shaped and defined as follows
119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)
where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866
We consider the following problem
Δ119906 = 0 in 119866
119906 = V (119903 120579) on 120574
(48)
where
V (119903 120579) = 11990323 sin(
2
3120579) + 00051119903
163 cos(16
3120579) (49)
is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as
119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω
1 (50)
where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then
119866119873119878
= 119866 119866119878 is a ldquononsingularrdquo part of 119866
The given domain 119866 is covered by four overlappingrectangles Π
119896 119896 = 1 4 and by the block sector 119879
3
1
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
paper is organized as follows in Section 2 the boundaryvalue problem and the integral representations of the exactsolution in each ldquosingularrdquo part are given In Section 3 tosupport the aim of the paper a Dirichlet problem on therectangle for the known exact solution from 119862
119896120582 119896 = 3 4 issolved using the 9-point scheme and the numerical results areillustrated In Section 4 the system of block-grid equationsand the convergence theorems are given In Section 5 a highlyaccurate approximation of the coefficient of the leadingsingular term of the exact solution (stress intensity factor) isgiven In Section 6 the method is illustrated for solving theproblem in L-shaped polygonwith the corner singularityTheconclusions are summarized in Section 7
2 Dirichlet Problem on a Staircase Polygon
Let 119866 be an open simply connected polygon 120574119895 119895 =
1 2 119873 its sides including the ends enumerated coun-terclockwise 120574 = 120574
1cup sdot sdot sdot cup 120574
119873the boundary of 119866 and 120572
119895120587
(120572119895= 12 1 32 2) the interior angle formed by the sides
120574119895minus1
and 120574119895 (120574
0= 120574
119873) Denote by 119860
119895= 120574
119895minus1cap 120574
119895the vertex of
the 119895th angle and by 119903119895 120579
119895a polar system of coordinates with a
pole in119860119895 where the angle 120579
119895is taken counterclockwise from
the side 120574119895
We consider the boundary value problem
Δ119906 = 0 on 119866 119906 = 120593119895(119904) on 120574
119895 1 le 119895 le 119873 (1)
whereΔ equiv 1205972120597119909
2+120597
2120597119910
2 120593
119895is a given function on 120574
119895of the
arc length 119904 taken along 120574 and 120593119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 that
is 120593119895has the fourth-order derivative on 120574
119895 which satisfies a
Holder condition with exponent 120582At some vertices119860
119895 (119904 = 119904
119895) for 120572
119895= 12 the conjugation
conditions
120593(2119902)
119895minus1(119904
119895) = (minus1)
119902120593(2119902)
119895(119904
119895) 119902 = 0 1 (2)
are fulfilled For the remaining vertices 119860119895 the values of 120593
119895minus1
and 120593119895at 119860
119895might be different Let 119864 be the set of all 119895 (1 le
119895 le 119873) for which 120572119895
= 12 or 120572119895= 12 but (2) is not fulfilled
In the neighborhood of 119860119895 119895 isin 119864 we construct two fixed
block sectors 119879119894
119895= 119879
119895(119903
119895119894) sub 119866 119894 = 1 2 where 0 lt 119903
1198952lt 119903
1198951lt
min119904119895+1
minus119904119895 119904
119895minus119904
119895minus1 119879
119895(119903) = (119903
119895 120579
119895) 0 lt 119903
119895lt 119903 0 lt 120579
119895lt
120572119895120587Let (see [10])
1205931198950(119905) = 120593
119895(119904
119895+ 119905) minus 120593
119895(119904
119895)
1205931198951(119905) = 120593
119895minus1(119904
119895minus 119905) minus 120593
119895minus1(119904
119895)
(3)
119876119895(119903
119895 120579
119895) = 120593
119895(119904
119895) +
(120593119895minus1
(119904119895) minus 120593
119895(119904
119895)) 120579
119895
120572119895120587
+1
120587
1
sum
119896=0
int
120590119895119896
0
119910119895120593119895119896
(119905120572119895) 119889119905
(119905 minus (minus1)119896119909119895)2
+ 1199102
119895
(4)
where
119909119895= 119903
1120572119895
119895cos(
120579119895
120572119895
) 119910119895= 119903
1120572119895
119895sin(
120579119895
120572119895
)
120590119895119896
=10038161003816100381610038161003816119904119895+1minus119896
minus 119904119895minus119896
10038161003816100381610038161003816
1120572119895
(5)
The function119876119895(119903
119895 120579
119895) is harmonic on119879
1
119895and satisfies the
boundary conditions in (1) on 120574119895minus1
cap 1198791
119895and 120574
119895cap 119879
1
119895 119895 isin 119864
except for the point 119860119895when 120593
119895minus1(119904
119895) = 120593
119895(119904
119895)
We formally set the value of 119876119895(119903
119895 120579
119895) and the solution
119906 of the problem (1) at the vertex 119860119895equal to (120593
119895minus1(119904
119895) +
120593119895(119904
119895))2 119895 isin 119864Let
119877119895(119903 120579 120578)
=1
120572119895
1
sum
119896=0
(minus1)119896119877((
119903
1199031198952
)
1120572119895
120579
120572119895
(minus1)119896 120578
120572119895
)
119895 isin 119864
(6)
where
119877 (119903 120579 120578) =1 minus 119903
2
2120587 (1 minus 2119903 cos (120579 minus 120578) + 1199032)(7)
is the kernel of the Poisson integral for a unit circle
Lemma 1 (see [10]) The solution 119906 of the boundary valueproblem (1) can be represented on 119879
2
119895 119881
119895 119895 isin 119864 in the form
119906 (119903119895 120579
119895) = 119876
119895(119903
119895 120579
119895)
+ int
120572119895120587
0
119877119895(119903
119895 120579
119895 120578) (119906 (119903
1198952 120578) minus 119876
119895(119903
1198952 120578)) 119889120578
(8)
where 119881119895is the curvilinear part of the boundary of 1198792
119895 and
119876119895(119903
119895 120579
119895) is the function defined by (4)
3 9-Point Solution on Rectangles
Let Π = (119909 119910) 0 lt 119909 lt 119886 0 lt 119910 lt 119887 be a rectanglewith 119886119887 being rational 120574
119895 119895 = 1 2 3 4 the sides including
the ends enumerated counterclockwise starting from the leftside (120574
0equiv 120574
4 120574
5equiv 120574
1) and 120574 = cup
4
119895=1120574119895the boundary of Π
We consider the boundary value problem
Δ119906 = 0 on Π
119906 = 120593119895
on 120574119895 119895 = 1 2 3 4
(9)
where 120593119895is the given function on 120574
119895
Definition 2 One says that the solution 119906 of the problem (9)belongs to 119862
4120582(Π) if
120593119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 119895 = 1 2 3 4 (10)
Abstract and Applied Analysis 3
and at the vertices 119860119895= 120574
119895minus1cap 120574
119895the conjugation conditions
120593(2119902)
119895= (minus1)
119902120593(2119902)
119895minus1 119902 = 0 1 (11)
are satisfied
Remark 3 From Theorem 31 in [11] it follows that the classof functions 1198624120582
(Π) is wider than 1198624120582
(Π)
Let ℎ gt 0 with 119886ℎ ge 2 119887ℎ ge 2 integers We assignto Π
ℎ a square net on Π with step ℎ obtained with the lines119910 = 0 ℎ 2ℎ Let 120574ℎ
119895be a set of nodes on the interior of 120574
119895
and let
120574ℎ
119895= 120574
119895cap 120574
119895+1 120574
ℎ= cup
4
119895=1(120574
ℎ
119895cup 120574
ℎ
119895)
Πℎ
= Πℎcup 120574
ℎ
(12)
We consider the system of finite difference equations
119906ℎ= 119861119906
ℎon Π
ℎ
119906ℎ= 120593
119895on 120574
ℎ
119895 119895 = 1 2 3 4
(13)
where
119861119906 (119909 119910)
equiv(119906 (119909 + ℎ 119910) + 119906 (119909 119910 + ℎ) + 119906 (119909 minus ℎ 119910) + 119906 (119909 119910 minus ℎ))
5
+ ( ((119906 (119909 + ℎ 119910 + ℎ) + 119906 (119909 minus ℎ 119910 + ℎ)
+ 119906 (119909 minus ℎ 119910 minus ℎ) + 119906 (119909 + ℎ 119910 minus ℎ)))
times 20minus1)
(14)
On the basis of the maximum principle the uniquesolvability of the system of finite difference equations (13)follows (see [12 Chapter 4])
Everywhere below we will denote constants which areindependent of ℎ and of the cofactors on their right by119888 119888
0 119888
1 generally using the same notation for different
constants for simplicity
Theorem 4 Let 119906 be the solution of problem (9) If 119906 isin
1198624120582
(Π) then
maxΠℎ
1003816100381610038161003816119906ℎ minus 1199061003816100381610038161003816 le ch4 (15)
where 119906ℎis the solution of the system (13)
Proof For the proof of this theorem see [13]
LetΠ1015840= (119909 119910) minus025 lt 119909 lt 025 0 lt 119910 lt 1 and let 1205741015840
be the boundary of Π1015840 We consider the Dirichlet problem
Δ119906 = 0 on Π1015840
119906 = V on 1205741015840
(16)
0 20 40 60 80 100 120 14011
12
13
14
15
16
17
119862400003
11986240003
1198624003
1198623055
1198623066
1198623075
119862308
Rϱ
hminus1 = 2ϱ
Figure 1 Dependence of the approximate solutions for the bound-ary functions from 119862
119896120582
where V = 119903119896+120582 cos(119896 + 120582)120579 119903 = radic1199092 + 1199102 0 lt 120582 lt 1 is the
exact solution of this problem which is from 119862119896120582
(Π1015840
)We solve the problem (16) by approximating 9-point
scheme when 119896 = 3 4 for the different values of 120582In Figure 1 the order of numerical convergence
R984858
Πℎ =
maxΠℎ10038161003816100381610038161199062minus984858 minus 119906
1003816100381610038161003816
maxΠℎ10038161003816100381610038161199062minus(984858+1) minus 119906
1003816100381610038161003816
(17)
of the 9-point solution 119906ℎ for different ℎ = 2
minus984858 and 984858 =
4 5 6 7 is demonstratedThese results show that the order ofnumerical convergence when the exact solution 119906 isin 119862
119896120582(Π)
depends on 119896 and 120582 and is119874(ℎ4)when 119896 = 4 which supports
estimation (15) Moreover this dependence also requires theuse of fourth-order gluing operator for all quadrature nodesin the construction of the system of block-grid equationswhen the given boundary functions are from the Holderclasses 1198624120582
4 System of Block-Grid Equations
In addition to the sectors 1198791
119895and 119879
2
119895(see Section 2) in the
neighborhood of each vertex 119860119895 119895 isin 119864 of the polygon 119866 we
construct two more sectors 1198793
119895and 119879
4
119895 where 0 lt 119903
1198954lt 119903
1198953lt
1199031198952 119903
1198953= (119903
1198952+ 119903
1198954)2 and 119879
3
119896cap 119879
3
119897= 0 119896 = 119897 119896 119897 isin 119864 and let
119866119879= 119866 (cup
119895isin1198641198794
119895)
We cover the given solution domain (a staircase polygon)by the finite number of sectors 119879
1
119895 119895 isin 119864 and rectangles
Π119896
sub 119866119879 119896 = 1 2 119872 as is shown in Figure 2 for
the case of 119871-shaped polygon where 119895 = 1119872 = 4 (seealso [2]) It is assumed that for the sides 119886
1119896and 119886
2119896of
Π119896the quotient 119886
1119896119886
2119896is rational and 119866 = (cup
119872
119896=1Π
119896) cup
(cup119895isin119864
1198793
119895) Let 120578
119896be the boundary of the rectangleΠ
119896 let119881
119895be
4 Abstract and Applied Analysis
minus1 minus05 0 05 1minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
Π1
Π212057810
12057820
119886
119887119888
119889119890
120579
1198601 1205741
1205740
11987941
11987931
11987921
11987911
Π3
Π4
12057820
12057830
12057830
12057840
119903
11990311
11990312
11990313 11990314
119866119878
Figure 2 Description of BGM for the L-shaped domain
the curvilinear part of the boundary of the sector 1198792
119895 and let
119905119896119895
= 120578119896cap 119879
3
119895 We choose a natural number 119899 and define
the quantities 119899(119895) = max4 [120572119895119899] 120573
119895= 120572
119895120587119899(119895) and
120579119898
119895= (119898 minus 12)120573
119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881
119895
we take the points (1199031198952 120579
119898
119895) 1 le 119898 le 119899(119895) and denote the
set of these points by 119881119899
119895 We introduce the parameter ℎ isin
(0 12060004] where 120600
0is a gluing depth of the rectanglesΠ
119896 119896 =
1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872
withmaximal possible step ℎ119896le minℎmin119886
1119896 119886
21198966 such
that the boundary 120578119896lies entirely on the grid lines Let Πℎ
119896be
the set of grid nodes on Π119896 let 120578ℎ
119896be the set of nodes on 120578
119896
and let Πℎ
119896= Π
ℎ
119896cup 120578
ℎ
119896 We denote the set of nodes on the
closure of 120578119896cap 119866
119879by 120578
ℎ
1198960 the set of nodes on 119905
119896119895by 119905
ℎ
119896119895 and
the set of remaining nodes on 120578119896by 120578
ℎ
1198961
Let
120596ℎ119899
= (cup119872
119896=1120578ℎ
1198960) cup (cup
119895isin119864119881
119899
119895)
119866ℎ119899
119879= 120596
ℎ119899cup (cup
119872
119896=1Π
ℎ
119896)
(18)
Let 120593 = 120593119895119873
119895=1 where 120593
119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 is
the given function in (1) We use the matching operator 1198784
at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906
ℎ 120593) at the point 119875 isin 120596
ℎ119899 is expressed linearly in termsof the values of 119906
ℎat the points 119875
119896of the grid constructed on
Π119896(119875)
(119875 isin Π119896(119875)
) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)
119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906
ℎ 0) has the representation
1198784(119906
ℎ 0) = sum
0le119897le15
120585119897119906ℎ119897 (19)
where 119906ℎ119896
= 119906ℎ(119875
119896)
120585119897ge 0 sum
0le119897le15
120585119897= 1 (20)
119906 minus 1198784(119906 120593) = 119874 (ℎ
4) (21)
Let 120596ℎ119899
119868sub 120596
ℎ119899 be the set of such points 119875 isin 120596ℎ119899
for which all points 119875119897in expression (19) are in cup
119872
119896=1Π
ℎ
119896 If
some points 119875119897in (19) emerge through the side 120574
119898 then the
set of such points 119875 is denoted by 120596ℎ119899
119863 According to the
construction of 1198784 in [14] the expression 1198784(119906
ℎ 120593) at each
point 119875 isin 120596ℎ119899
= 120596ℎ119899
119868cup 120596
ℎ119899
119863can be expressed as follows
1198784(119906
ℎ 120593)
=
1198784119906ℎ 119875 isin 120596
ℎ119899
119868
1198784(119906
ℎminus
3
sum
119896=0
119886119896Re 119911119896) + (
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(22)
where
119886119896=
1
119896
119889119896120593119898(119904)
119889119904119896
100381610038161003816100381610038161003816100381610038161003816119904=119904119875
119896 = 0 1 2 3 (23)
and 119904119875corresponds to such a point119876 isin 120574
119898for which the line
119875119876 is perpendicular to 120574119898
Let
119876119895= 119876
119895(119903
119895 120579
119895) 119876
119902
1198952= 119876
119895(119903
1198952 120579
119902
119895) (24)
The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876
120576
119895and 119876
119902120576
1198952of the
quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876
120576
119895minus 119876
119895
10038161003816100381610038161003816le 119888
1120576
10038161003816100381610038161003816119876
119902120576
1198952minus 119876
119902
1198952
10038161003816100381610038161003816le 119888
2120576 (25)
where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888
2are constants
independent of 120576Consider the system of linear algebraic equations
119906120576
ℎ= 119861119906
120576
ℎon Π
ℎ
119896
119906120576
ℎ= 120593
119898on 120578
ℎ
1198961cap 120574
119898
119906120576
ℎ(119903
119895 120579
119895)
= 119876120576
119895+ 120573
119895
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
times 119877119895(119903
119895 120579
119895 120579
119902
119895) on (119903
119895 120579
119895) isin 119905
ℎ
119896119895
119906120576
ℎ= 119878
4119906120576
ℎon 120596
ℎ119899
(26)
where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864
Abstract and Applied Analysis 5
Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2
minus984858 and 120576 = 5 times 10minus13
(2minus984858 119899) 120577
120576
ℎ119866119873119878 R
984858
119866119873119878
(2minus4 60) 1609 times 10
minus815577
(2minus5 170) 1033 times 10
minus9
(2minus5 130) 1191 times 10
minus916690
(2minus6 150) 7136 times 10
minus11
(2minus5 140) 1136 times 10
minus916259
(2minus6 170) 6991 times 10
minus11
(2minus6 100) 2169 times 10
minus1017096
(2minus7 130) 1269 times 10
minus11
Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =
2minus984858 and 120576 = 5 times 10
minus13
(2minus984858 119899) 120577
120576
ℎ119866119878 R
984858
119866119878
(2minus4 100) 1931 times 10
minus816078
(2minus5 150) 1182 times 10
minus9
(2minus5 130) 1294 times 10
minus916789
(2minus6 150) 7708 times 10
minus11
(2minus5 140) 1312 times 10
minus917967
(2minus6 170) 7304 times 10
minus11
(2minus6 100) 2389 times 10
minus1018164
(2minus7 130) 1315 times 10
minus11
Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) 120577
120576
ℎ119866119873119878 120577
120576
ℎ119866119878 Iteration
(16 70) 1139 times 10minus8
1572 times 10minus8
22
(32 170) 1033 times 10minus9
1184 times 10minus9
23
(64 170) 6990 times 10minus11
7304 times 10minus11
24
(128 200) 8628 times 10minus12
8833 times 10minus12
25
Let 119879lowast
119895= 119879
119895(119903
lowast
119895) be the sector where 119903lowast
119895= (119903
1198952+119903
1198953)2 119895 isin
119864 and let 119906120576ℎ(119903
1198952 120579
119902
119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution
values of the system (26) on 119881ℎ
119895(at the quadrature nodes)
The function
119880120576
ℎ(119903
119895 120579
119895) = 119876
119895(119903
119895 120579
119895)
+ 120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) (119906
120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
(27)
defined on 119879lowast
119895 is called an approximate solution of the
problem (1) on the closed block 1198793
119895 119895 isin 119864
Definition 5 The system (26) and (27) is called the system ofblock-grid equations
Theorem 6 There is a natural number 1198990 such that for all
119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution
Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899
0and 120590 such that for all 119899 ge 119899
0
max(119903119895120579119895)isin119879
3
119895
120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) le 120590 lt 1 (28)
The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)
Theorem 7 There exists a natural number 1198990 such that for all
119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)
where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid
max119866ℎ119899
119879
1003816100381610038161003816119906120576
ℎminus 119906
1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816le 119888
119901(ℎ
4+ 120576) on 119879
3
119895
(31)
for integer 1120572119895when 119901 ge 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895
(32)
for any 1120572119895 if 0 le 119901 lt 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895 119860
119895
(33)
for noninteger 1120572119895 when 119901 gt 1120572
119895 Everywhere 0 le 119902 le 119901 119906
is the exact solution of the problem (1) and119880120576
ℎ(119903
119895 120579
119895) is defined
by formula (27)
Proof Let
120585120576
ℎ= 119906
120576
ℎminus 119906 (34)
where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866
ℎ119899
119879
of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576
ℎsatisfies the system of difference equations
120585120576
ℎ= 119861120585
120576
ℎ+ 119903
1
ℎon Π
ℎ
119896
120585120576
ℎ= 0 on 120578
ℎ
1198961
120585120576
ℎ(119903
119895 120579
119895) = 120573
119895
119899(119895)
sum
119902=1
120585120576
ℎ(119903
1198952 120579
119902
119895) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
+ 1199032
119895ℎ (119903
119895 120579
119895) isin 119905
ℎ
119896119895
120585120576
ℎ= 119878
4120585120576
ℎ+ 119903
3
ℎon 120596
ℎ119899
(35)
6 Abstract and Applied Analysis
Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 3895 times 10minus7
3895 times 10minus7
(32 170) 4627 times 10minus8
4627 times 10minus8
(64 170) 1124 times 10minus9
3125 times 10minus9
(128 200) 2214 times 10minus10
2233 times 10minus10
Table 5 In119866119878 the minimum errors of the derivatives over the pairs
(ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 9663 times 10minus6
9663 times 10minus6
(32 170) 9653 times 10minus6
9653 times 10minus6
(64 170) 9649 times 10minus6
9649 times 10minus6
(128 200) 9648 times 10minus6
9648 times 10minus6
where 1 le 119896 le 119872 119895 isin 119864
1199031
ℎ= 119861119906 minus 119906 on cup
119872
119896=1Π
ℎ
119896 (36)
1199032
119895ℎ= 120573
119895
119899(119895)
sum
119902=1
(119906 (1199031198952 120579
119902
119895) minus 119876
119902120576
1198952) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
minus (119906 minus 119876120576
119895) on cup
119872
119896=1(cup
119895isin119864119905ℎ
119896119895)
(37)
1199033
ℎ
=
1198784119906 minus 119906 on 120596
ℎ119899
119868
1198784(119906 minus
3
sum
119896=0
119886119896Re 119911119896) minus (119906 minus
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(38)
On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows
The function 119880120576
ℎ(119903
119895 120579
119895) given by formula (27) defined
on the closed sector 119879lowast
119895 119895 isin 119864 where 119903
lowast
119895= (119903
1198952+ 119903
1198953)2
and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879
2
119895 119881
119895 119895 isin 119864 and then the
difference function 120577120576
ℎ(119903
119895 120579
119895) = 119880
120576
ℎ(119903
119895 120579
119895)minus119906(119903
119895 120579
119895) is defined
on 119879lowast
119895 119895 isin 119864 and
120577120576
ℎ(119903
lowast
119895 0) = 120577
120576
ℎ(119903
lowast
119895 120572
119895120587) = 0 119895 isin 119864 (39)
On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge
max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain
10038161003816100381610038161003816120577120576
ℎ(119903
119895 120579
119895)10038161003816100381610038161003816le 119888 (ℎ
4+ 120576) on 119879
lowast
119895 119895 isin 119864 (40)
Furthermore the function 120577120576
ℎ(119903
119895 120579
119895) continuous on 119879
lowast
119895is a
solution of the following Dirichlet problem
Δ120577120576
ℎ= 0 on 119879
lowast
119895
120577120576
ℎ= 0 on 120574
119898cap 119879
lowast
119895 119898 = 119895 minus 1 119895
120577120576
ℎ(119903
lowast
119895 120579
119895) = 119880
120576
ℎ(119903
lowast
119895 120579
119895) minus 119906 (119903
lowast
119895 120579
119895) 0 le 120579
119895le 120572
119895120587
(41)
Since 1198793
119895sub 119879
lowast
119895 on the basis of (39) and (40) from Lemma
612 in [16] inequalities (31)ndash(33) of Theorem 7 follow
5 Stress Intensity Factor
Let in the condition 120593119895isin 119862
4120582(120574
119895) the exponent 120582 be such
that
120572119895(4 + 120582) = 0 2120572
119895(4 + 120582) = 0 (42)
where sdot is the symbol of fractional partThese conditions forthe given 120572
119895can be fulfilled by decreasing 120582
On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879
lowast
119895 119895 isin 119864 as follows
119906 (119909119895 119910
119895) = (119909
119895 119910
119895) +
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911
+
119899120572119895
sum
119896=1
120591(119895)
119896119903119896120572119895
119895sin
119896120579119895
120572119895
(43)
where 119899120572119895
= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909
119895+119894119910
119895 120583(119895)
119896
and 120591(119895)
119896are some numbers and (119909
119895 119910
119895) isin 119862
4120582(119879
2
119895) is the
harmonic on 1198792
119895 By taking 120579
119895= 120572
1198951205872 from the formula
(43) it follows that the coefficient 120591(119895)1
which is called the stressintensity factor can be represented as
120591(119895)
1= lim
119903119895rarr0
1
1199031120572119895
119895
(119906 (119909119895 119910
119895) minus (119909
119895 119910
119895)
minus
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911)
(44)
Abstract and Applied Analysis 7
Table 6 The stress intensity factor 1205911205981119899
for 119899 = 70 170 200 when 120576 = 5 times 10minus13
ℎminus1
120591120576
170120591120576
1170120591120576
1200
16 1000000014856688 1000000017180415 1000000017197438
32 1000000005800844 1000000001230267 1000000001236709
64 1000000004138169 1000000000073107 1000000000079938
128 1000000004053153 1000000000003531 1000000000003267
10minus15 10minus10 10minus5 10010minus10
10minus5
100
120576
Max
erro
r
ℎ = 116 119899 = 70
(a)
10minus10
10minus5
100
10minus15 10minus10 10minus5 100
120576
Max
erro
r
ℎ = 132 119899 = 170
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 3 Dependence on 120576 for ℎminus1 = 16 32
From formula (44) it follows that 120591(119895)
1can be approxi-
mated by
120591(119895)120576
1119899
= lim119903119895rarr0
1
1199031120572119895
119895
(119880120576
ℎ(119903
119895 120579
119895)
minus(120593119895(119904
119895) + (120593
119895minus1(119904
119895) minus 120593
119895(119904
119895))
120579119895
120572119895120587))
(45)
Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula
120591(119895)120576
1119899=
1
120587int
1205901198950
0
1205931198950(119905
120572119895) 119889119905
1199052+
1
120587int
1205901198951
0
1205931198951(119905
120572119895) 119889119905
1199052
+2
119899 (119895) 1199031120572119895
1198952
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952) sin 1
120572119895
120579119902
119895
(46)
This formula is obtained for the second-order BGM in [8]
10minus15 10minus10 10minus5 100
120576
h = 164 n = 170100
10minus10
10minus20
Max
erro
r
(a)
10minus15 10minus10 10minus5 100
120576
h = 1128 n = 200100
10minus10
10minus20
Max
erro
r
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 4 Dependence on 120576 for ℎminus1 = 64 128
6 Numerical Results
Let 119866 be L-shaped and defined as follows
119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)
where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866
We consider the following problem
Δ119906 = 0 in 119866
119906 = V (119903 120579) on 120574
(48)
where
V (119903 120579) = 11990323 sin(
2
3120579) + 00051119903
163 cos(16
3120579) (49)
is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as
119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω
1 (50)
where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then
119866119873119878
= 119866 119866119878 is a ldquononsingularrdquo part of 119866
The given domain 119866 is covered by four overlappingrectangles Π
119896 119896 = 1 4 and by the block sector 119879
3
1
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
and at the vertices 119860119895= 120574
119895minus1cap 120574
119895the conjugation conditions
120593(2119902)
119895= (minus1)
119902120593(2119902)
119895minus1 119902 = 0 1 (11)
are satisfied
Remark 3 From Theorem 31 in [11] it follows that the classof functions 1198624120582
(Π) is wider than 1198624120582
(Π)
Let ℎ gt 0 with 119886ℎ ge 2 119887ℎ ge 2 integers We assignto Π
ℎ a square net on Π with step ℎ obtained with the lines119910 = 0 ℎ 2ℎ Let 120574ℎ
119895be a set of nodes on the interior of 120574
119895
and let
120574ℎ
119895= 120574
119895cap 120574
119895+1 120574
ℎ= cup
4
119895=1(120574
ℎ
119895cup 120574
ℎ
119895)
Πℎ
= Πℎcup 120574
ℎ
(12)
We consider the system of finite difference equations
119906ℎ= 119861119906
ℎon Π
ℎ
119906ℎ= 120593
119895on 120574
ℎ
119895 119895 = 1 2 3 4
(13)
where
119861119906 (119909 119910)
equiv(119906 (119909 + ℎ 119910) + 119906 (119909 119910 + ℎ) + 119906 (119909 minus ℎ 119910) + 119906 (119909 119910 minus ℎ))
5
+ ( ((119906 (119909 + ℎ 119910 + ℎ) + 119906 (119909 minus ℎ 119910 + ℎ)
+ 119906 (119909 minus ℎ 119910 minus ℎ) + 119906 (119909 + ℎ 119910 minus ℎ)))
times 20minus1)
(14)
On the basis of the maximum principle the uniquesolvability of the system of finite difference equations (13)follows (see [12 Chapter 4])
Everywhere below we will denote constants which areindependent of ℎ and of the cofactors on their right by119888 119888
0 119888
1 generally using the same notation for different
constants for simplicity
Theorem 4 Let 119906 be the solution of problem (9) If 119906 isin
1198624120582
(Π) then
maxΠℎ
1003816100381610038161003816119906ℎ minus 1199061003816100381610038161003816 le ch4 (15)
where 119906ℎis the solution of the system (13)
Proof For the proof of this theorem see [13]
LetΠ1015840= (119909 119910) minus025 lt 119909 lt 025 0 lt 119910 lt 1 and let 1205741015840
be the boundary of Π1015840 We consider the Dirichlet problem
Δ119906 = 0 on Π1015840
119906 = V on 1205741015840
(16)
0 20 40 60 80 100 120 14011
12
13
14
15
16
17
119862400003
11986240003
1198624003
1198623055
1198623066
1198623075
119862308
Rϱ
hminus1 = 2ϱ
Figure 1 Dependence of the approximate solutions for the bound-ary functions from 119862
119896120582
where V = 119903119896+120582 cos(119896 + 120582)120579 119903 = radic1199092 + 1199102 0 lt 120582 lt 1 is the
exact solution of this problem which is from 119862119896120582
(Π1015840
)We solve the problem (16) by approximating 9-point
scheme when 119896 = 3 4 for the different values of 120582In Figure 1 the order of numerical convergence
R984858
Πℎ =
maxΠℎ10038161003816100381610038161199062minus984858 minus 119906
1003816100381610038161003816
maxΠℎ10038161003816100381610038161199062minus(984858+1) minus 119906
1003816100381610038161003816
(17)
of the 9-point solution 119906ℎ for different ℎ = 2
minus984858 and 984858 =
4 5 6 7 is demonstratedThese results show that the order ofnumerical convergence when the exact solution 119906 isin 119862
119896120582(Π)
depends on 119896 and 120582 and is119874(ℎ4)when 119896 = 4 which supports
estimation (15) Moreover this dependence also requires theuse of fourth-order gluing operator for all quadrature nodesin the construction of the system of block-grid equationswhen the given boundary functions are from the Holderclasses 1198624120582
4 System of Block-Grid Equations
In addition to the sectors 1198791
119895and 119879
2
119895(see Section 2) in the
neighborhood of each vertex 119860119895 119895 isin 119864 of the polygon 119866 we
construct two more sectors 1198793
119895and 119879
4
119895 where 0 lt 119903
1198954lt 119903
1198953lt
1199031198952 119903
1198953= (119903
1198952+ 119903
1198954)2 and 119879
3
119896cap 119879
3
119897= 0 119896 = 119897 119896 119897 isin 119864 and let
119866119879= 119866 (cup
119895isin1198641198794
119895)
We cover the given solution domain (a staircase polygon)by the finite number of sectors 119879
1
119895 119895 isin 119864 and rectangles
Π119896
sub 119866119879 119896 = 1 2 119872 as is shown in Figure 2 for
the case of 119871-shaped polygon where 119895 = 1119872 = 4 (seealso [2]) It is assumed that for the sides 119886
1119896and 119886
2119896of
Π119896the quotient 119886
1119896119886
2119896is rational and 119866 = (cup
119872
119896=1Π
119896) cup
(cup119895isin119864
1198793
119895) Let 120578
119896be the boundary of the rectangleΠ
119896 let119881
119895be
4 Abstract and Applied Analysis
minus1 minus05 0 05 1minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
Π1
Π212057810
12057820
119886
119887119888
119889119890
120579
1198601 1205741
1205740
11987941
11987931
11987921
11987911
Π3
Π4
12057820
12057830
12057830
12057840
119903
11990311
11990312
11990313 11990314
119866119878
Figure 2 Description of BGM for the L-shaped domain
the curvilinear part of the boundary of the sector 1198792
119895 and let
119905119896119895
= 120578119896cap 119879
3
119895 We choose a natural number 119899 and define
the quantities 119899(119895) = max4 [120572119895119899] 120573
119895= 120572
119895120587119899(119895) and
120579119898
119895= (119898 minus 12)120573
119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881
119895
we take the points (1199031198952 120579
119898
119895) 1 le 119898 le 119899(119895) and denote the
set of these points by 119881119899
119895 We introduce the parameter ℎ isin
(0 12060004] where 120600
0is a gluing depth of the rectanglesΠ
119896 119896 =
1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872
withmaximal possible step ℎ119896le minℎmin119886
1119896 119886
21198966 such
that the boundary 120578119896lies entirely on the grid lines Let Πℎ
119896be
the set of grid nodes on Π119896 let 120578ℎ
119896be the set of nodes on 120578
119896
and let Πℎ
119896= Π
ℎ
119896cup 120578
ℎ
119896 We denote the set of nodes on the
closure of 120578119896cap 119866
119879by 120578
ℎ
1198960 the set of nodes on 119905
119896119895by 119905
ℎ
119896119895 and
the set of remaining nodes on 120578119896by 120578
ℎ
1198961
Let
120596ℎ119899
= (cup119872
119896=1120578ℎ
1198960) cup (cup
119895isin119864119881
119899
119895)
119866ℎ119899
119879= 120596
ℎ119899cup (cup
119872
119896=1Π
ℎ
119896)
(18)
Let 120593 = 120593119895119873
119895=1 where 120593
119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 is
the given function in (1) We use the matching operator 1198784
at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906
ℎ 120593) at the point 119875 isin 120596
ℎ119899 is expressed linearly in termsof the values of 119906
ℎat the points 119875
119896of the grid constructed on
Π119896(119875)
(119875 isin Π119896(119875)
) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)
119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906
ℎ 0) has the representation
1198784(119906
ℎ 0) = sum
0le119897le15
120585119897119906ℎ119897 (19)
where 119906ℎ119896
= 119906ℎ(119875
119896)
120585119897ge 0 sum
0le119897le15
120585119897= 1 (20)
119906 minus 1198784(119906 120593) = 119874 (ℎ
4) (21)
Let 120596ℎ119899
119868sub 120596
ℎ119899 be the set of such points 119875 isin 120596ℎ119899
for which all points 119875119897in expression (19) are in cup
119872
119896=1Π
ℎ
119896 If
some points 119875119897in (19) emerge through the side 120574
119898 then the
set of such points 119875 is denoted by 120596ℎ119899
119863 According to the
construction of 1198784 in [14] the expression 1198784(119906
ℎ 120593) at each
point 119875 isin 120596ℎ119899
= 120596ℎ119899
119868cup 120596
ℎ119899
119863can be expressed as follows
1198784(119906
ℎ 120593)
=
1198784119906ℎ 119875 isin 120596
ℎ119899
119868
1198784(119906
ℎminus
3
sum
119896=0
119886119896Re 119911119896) + (
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(22)
where
119886119896=
1
119896
119889119896120593119898(119904)
119889119904119896
100381610038161003816100381610038161003816100381610038161003816119904=119904119875
119896 = 0 1 2 3 (23)
and 119904119875corresponds to such a point119876 isin 120574
119898for which the line
119875119876 is perpendicular to 120574119898
Let
119876119895= 119876
119895(119903
119895 120579
119895) 119876
119902
1198952= 119876
119895(119903
1198952 120579
119902
119895) (24)
The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876
120576
119895and 119876
119902120576
1198952of the
quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876
120576
119895minus 119876
119895
10038161003816100381610038161003816le 119888
1120576
10038161003816100381610038161003816119876
119902120576
1198952minus 119876
119902
1198952
10038161003816100381610038161003816le 119888
2120576 (25)
where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888
2are constants
independent of 120576Consider the system of linear algebraic equations
119906120576
ℎ= 119861119906
120576
ℎon Π
ℎ
119896
119906120576
ℎ= 120593
119898on 120578
ℎ
1198961cap 120574
119898
119906120576
ℎ(119903
119895 120579
119895)
= 119876120576
119895+ 120573
119895
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
times 119877119895(119903
119895 120579
119895 120579
119902
119895) on (119903
119895 120579
119895) isin 119905
ℎ
119896119895
119906120576
ℎ= 119878
4119906120576
ℎon 120596
ℎ119899
(26)
where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864
Abstract and Applied Analysis 5
Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2
minus984858 and 120576 = 5 times 10minus13
(2minus984858 119899) 120577
120576
ℎ119866119873119878 R
984858
119866119873119878
(2minus4 60) 1609 times 10
minus815577
(2minus5 170) 1033 times 10
minus9
(2minus5 130) 1191 times 10
minus916690
(2minus6 150) 7136 times 10
minus11
(2minus5 140) 1136 times 10
minus916259
(2minus6 170) 6991 times 10
minus11
(2minus6 100) 2169 times 10
minus1017096
(2minus7 130) 1269 times 10
minus11
Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =
2minus984858 and 120576 = 5 times 10
minus13
(2minus984858 119899) 120577
120576
ℎ119866119878 R
984858
119866119878
(2minus4 100) 1931 times 10
minus816078
(2minus5 150) 1182 times 10
minus9
(2minus5 130) 1294 times 10
minus916789
(2minus6 150) 7708 times 10
minus11
(2minus5 140) 1312 times 10
minus917967
(2minus6 170) 7304 times 10
minus11
(2minus6 100) 2389 times 10
minus1018164
(2minus7 130) 1315 times 10
minus11
Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) 120577
120576
ℎ119866119873119878 120577
120576
ℎ119866119878 Iteration
(16 70) 1139 times 10minus8
1572 times 10minus8
22
(32 170) 1033 times 10minus9
1184 times 10minus9
23
(64 170) 6990 times 10minus11
7304 times 10minus11
24
(128 200) 8628 times 10minus12
8833 times 10minus12
25
Let 119879lowast
119895= 119879
119895(119903
lowast
119895) be the sector where 119903lowast
119895= (119903
1198952+119903
1198953)2 119895 isin
119864 and let 119906120576ℎ(119903
1198952 120579
119902
119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution
values of the system (26) on 119881ℎ
119895(at the quadrature nodes)
The function
119880120576
ℎ(119903
119895 120579
119895) = 119876
119895(119903
119895 120579
119895)
+ 120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) (119906
120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
(27)
defined on 119879lowast
119895 is called an approximate solution of the
problem (1) on the closed block 1198793
119895 119895 isin 119864
Definition 5 The system (26) and (27) is called the system ofblock-grid equations
Theorem 6 There is a natural number 1198990 such that for all
119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution
Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899
0and 120590 such that for all 119899 ge 119899
0
max(119903119895120579119895)isin119879
3
119895
120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) le 120590 lt 1 (28)
The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)
Theorem 7 There exists a natural number 1198990 such that for all
119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)
where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid
max119866ℎ119899
119879
1003816100381610038161003816119906120576
ℎminus 119906
1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816le 119888
119901(ℎ
4+ 120576) on 119879
3
119895
(31)
for integer 1120572119895when 119901 ge 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895
(32)
for any 1120572119895 if 0 le 119901 lt 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895 119860
119895
(33)
for noninteger 1120572119895 when 119901 gt 1120572
119895 Everywhere 0 le 119902 le 119901 119906
is the exact solution of the problem (1) and119880120576
ℎ(119903
119895 120579
119895) is defined
by formula (27)
Proof Let
120585120576
ℎ= 119906
120576
ℎminus 119906 (34)
where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866
ℎ119899
119879
of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576
ℎsatisfies the system of difference equations
120585120576
ℎ= 119861120585
120576
ℎ+ 119903
1
ℎon Π
ℎ
119896
120585120576
ℎ= 0 on 120578
ℎ
1198961
120585120576
ℎ(119903
119895 120579
119895) = 120573
119895
119899(119895)
sum
119902=1
120585120576
ℎ(119903
1198952 120579
119902
119895) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
+ 1199032
119895ℎ (119903
119895 120579
119895) isin 119905
ℎ
119896119895
120585120576
ℎ= 119878
4120585120576
ℎ+ 119903
3
ℎon 120596
ℎ119899
(35)
6 Abstract and Applied Analysis
Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 3895 times 10minus7
3895 times 10minus7
(32 170) 4627 times 10minus8
4627 times 10minus8
(64 170) 1124 times 10minus9
3125 times 10minus9
(128 200) 2214 times 10minus10
2233 times 10minus10
Table 5 In119866119878 the minimum errors of the derivatives over the pairs
(ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 9663 times 10minus6
9663 times 10minus6
(32 170) 9653 times 10minus6
9653 times 10minus6
(64 170) 9649 times 10minus6
9649 times 10minus6
(128 200) 9648 times 10minus6
9648 times 10minus6
where 1 le 119896 le 119872 119895 isin 119864
1199031
ℎ= 119861119906 minus 119906 on cup
119872
119896=1Π
ℎ
119896 (36)
1199032
119895ℎ= 120573
119895
119899(119895)
sum
119902=1
(119906 (1199031198952 120579
119902
119895) minus 119876
119902120576
1198952) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
minus (119906 minus 119876120576
119895) on cup
119872
119896=1(cup
119895isin119864119905ℎ
119896119895)
(37)
1199033
ℎ
=
1198784119906 minus 119906 on 120596
ℎ119899
119868
1198784(119906 minus
3
sum
119896=0
119886119896Re 119911119896) minus (119906 minus
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(38)
On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows
The function 119880120576
ℎ(119903
119895 120579
119895) given by formula (27) defined
on the closed sector 119879lowast
119895 119895 isin 119864 where 119903
lowast
119895= (119903
1198952+ 119903
1198953)2
and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879
2
119895 119881
119895 119895 isin 119864 and then the
difference function 120577120576
ℎ(119903
119895 120579
119895) = 119880
120576
ℎ(119903
119895 120579
119895)minus119906(119903
119895 120579
119895) is defined
on 119879lowast
119895 119895 isin 119864 and
120577120576
ℎ(119903
lowast
119895 0) = 120577
120576
ℎ(119903
lowast
119895 120572
119895120587) = 0 119895 isin 119864 (39)
On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge
max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain
10038161003816100381610038161003816120577120576
ℎ(119903
119895 120579
119895)10038161003816100381610038161003816le 119888 (ℎ
4+ 120576) on 119879
lowast
119895 119895 isin 119864 (40)
Furthermore the function 120577120576
ℎ(119903
119895 120579
119895) continuous on 119879
lowast
119895is a
solution of the following Dirichlet problem
Δ120577120576
ℎ= 0 on 119879
lowast
119895
120577120576
ℎ= 0 on 120574
119898cap 119879
lowast
119895 119898 = 119895 minus 1 119895
120577120576
ℎ(119903
lowast
119895 120579
119895) = 119880
120576
ℎ(119903
lowast
119895 120579
119895) minus 119906 (119903
lowast
119895 120579
119895) 0 le 120579
119895le 120572
119895120587
(41)
Since 1198793
119895sub 119879
lowast
119895 on the basis of (39) and (40) from Lemma
612 in [16] inequalities (31)ndash(33) of Theorem 7 follow
5 Stress Intensity Factor
Let in the condition 120593119895isin 119862
4120582(120574
119895) the exponent 120582 be such
that
120572119895(4 + 120582) = 0 2120572
119895(4 + 120582) = 0 (42)
where sdot is the symbol of fractional partThese conditions forthe given 120572
119895can be fulfilled by decreasing 120582
On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879
lowast
119895 119895 isin 119864 as follows
119906 (119909119895 119910
119895) = (119909
119895 119910
119895) +
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911
+
119899120572119895
sum
119896=1
120591(119895)
119896119903119896120572119895
119895sin
119896120579119895
120572119895
(43)
where 119899120572119895
= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909
119895+119894119910
119895 120583(119895)
119896
and 120591(119895)
119896are some numbers and (119909
119895 119910
119895) isin 119862
4120582(119879
2
119895) is the
harmonic on 1198792
119895 By taking 120579
119895= 120572
1198951205872 from the formula
(43) it follows that the coefficient 120591(119895)1
which is called the stressintensity factor can be represented as
120591(119895)
1= lim
119903119895rarr0
1
1199031120572119895
119895
(119906 (119909119895 119910
119895) minus (119909
119895 119910
119895)
minus
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911)
(44)
Abstract and Applied Analysis 7
Table 6 The stress intensity factor 1205911205981119899
for 119899 = 70 170 200 when 120576 = 5 times 10minus13
ℎminus1
120591120576
170120591120576
1170120591120576
1200
16 1000000014856688 1000000017180415 1000000017197438
32 1000000005800844 1000000001230267 1000000001236709
64 1000000004138169 1000000000073107 1000000000079938
128 1000000004053153 1000000000003531 1000000000003267
10minus15 10minus10 10minus5 10010minus10
10minus5
100
120576
Max
erro
r
ℎ = 116 119899 = 70
(a)
10minus10
10minus5
100
10minus15 10minus10 10minus5 100
120576
Max
erro
r
ℎ = 132 119899 = 170
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 3 Dependence on 120576 for ℎminus1 = 16 32
From formula (44) it follows that 120591(119895)
1can be approxi-
mated by
120591(119895)120576
1119899
= lim119903119895rarr0
1
1199031120572119895
119895
(119880120576
ℎ(119903
119895 120579
119895)
minus(120593119895(119904
119895) + (120593
119895minus1(119904
119895) minus 120593
119895(119904
119895))
120579119895
120572119895120587))
(45)
Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula
120591(119895)120576
1119899=
1
120587int
1205901198950
0
1205931198950(119905
120572119895) 119889119905
1199052+
1
120587int
1205901198951
0
1205931198951(119905
120572119895) 119889119905
1199052
+2
119899 (119895) 1199031120572119895
1198952
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952) sin 1
120572119895
120579119902
119895
(46)
This formula is obtained for the second-order BGM in [8]
10minus15 10minus10 10minus5 100
120576
h = 164 n = 170100
10minus10
10minus20
Max
erro
r
(a)
10minus15 10minus10 10minus5 100
120576
h = 1128 n = 200100
10minus10
10minus20
Max
erro
r
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 4 Dependence on 120576 for ℎminus1 = 64 128
6 Numerical Results
Let 119866 be L-shaped and defined as follows
119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)
where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866
We consider the following problem
Δ119906 = 0 in 119866
119906 = V (119903 120579) on 120574
(48)
where
V (119903 120579) = 11990323 sin(
2
3120579) + 00051119903
163 cos(16
3120579) (49)
is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as
119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω
1 (50)
where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then
119866119873119878
= 119866 119866119878 is a ldquononsingularrdquo part of 119866
The given domain 119866 is covered by four overlappingrectangles Π
119896 119896 = 1 4 and by the block sector 119879
3
1
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
minus1 minus05 0 05 1minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
Π1
Π212057810
12057820
119886
119887119888
119889119890
120579
1198601 1205741
1205740
11987941
11987931
11987921
11987911
Π3
Π4
12057820
12057830
12057830
12057840
119903
11990311
11990312
11990313 11990314
119866119878
Figure 2 Description of BGM for the L-shaped domain
the curvilinear part of the boundary of the sector 1198792
119895 and let
119905119896119895
= 120578119896cap 119879
3
119895 We choose a natural number 119899 and define
the quantities 119899(119895) = max4 [120572119895119899] 120573
119895= 120572
119895120587119899(119895) and
120579119898
119895= (119898 minus 12)120573
119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881
119895
we take the points (1199031198952 120579
119898
119895) 1 le 119898 le 119899(119895) and denote the
set of these points by 119881119899
119895 We introduce the parameter ℎ isin
(0 12060004] where 120600
0is a gluing depth of the rectanglesΠ
119896 119896 =
1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872
withmaximal possible step ℎ119896le minℎmin119886
1119896 119886
21198966 such
that the boundary 120578119896lies entirely on the grid lines Let Πℎ
119896be
the set of grid nodes on Π119896 let 120578ℎ
119896be the set of nodes on 120578
119896
and let Πℎ
119896= Π
ℎ
119896cup 120578
ℎ
119896 We denote the set of nodes on the
closure of 120578119896cap 119866
119879by 120578
ℎ
1198960 the set of nodes on 119905
119896119895by 119905
ℎ
119896119895 and
the set of remaining nodes on 120578119896by 120578
ℎ
1198961
Let
120596ℎ119899
= (cup119872
119896=1120578ℎ
1198960) cup (cup
119895isin119864119881
119899
119895)
119866ℎ119899
119879= 120596
ℎ119899cup (cup
119872
119896=1Π
ℎ
119896)
(18)
Let 120593 = 120593119895119873
119895=1 where 120593
119895isin 119862
4120582(120574
119895) 0 lt 120582 lt 1 is
the given function in (1) We use the matching operator 1198784
at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906
ℎ 120593) at the point 119875 isin 120596
ℎ119899 is expressed linearly in termsof the values of 119906
ℎat the points 119875
119896of the grid constructed on
Π119896(119875)
(119875 isin Π119896(119875)
) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)
119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906
ℎ 0) has the representation
1198784(119906
ℎ 0) = sum
0le119897le15
120585119897119906ℎ119897 (19)
where 119906ℎ119896
= 119906ℎ(119875
119896)
120585119897ge 0 sum
0le119897le15
120585119897= 1 (20)
119906 minus 1198784(119906 120593) = 119874 (ℎ
4) (21)
Let 120596ℎ119899
119868sub 120596
ℎ119899 be the set of such points 119875 isin 120596ℎ119899
for which all points 119875119897in expression (19) are in cup
119872
119896=1Π
ℎ
119896 If
some points 119875119897in (19) emerge through the side 120574
119898 then the
set of such points 119875 is denoted by 120596ℎ119899
119863 According to the
construction of 1198784 in [14] the expression 1198784(119906
ℎ 120593) at each
point 119875 isin 120596ℎ119899
= 120596ℎ119899
119868cup 120596
ℎ119899
119863can be expressed as follows
1198784(119906
ℎ 120593)
=
1198784119906ℎ 119875 isin 120596
ℎ119899
119868
1198784(119906
ℎminus
3
sum
119896=0
119886119896Re 119911119896) + (
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(22)
where
119886119896=
1
119896
119889119896120593119898(119904)
119889119904119896
100381610038161003816100381610038161003816100381610038161003816119904=119904119875
119896 = 0 1 2 3 (23)
and 119904119875corresponds to such a point119876 isin 120574
119898for which the line
119875119876 is perpendicular to 120574119898
Let
119876119895= 119876
119895(119903
119895 120579
119895) 119876
119902
1198952= 119876
119895(119903
1198952 120579
119902
119895) (24)
The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876
120576
119895and 119876
119902120576
1198952of the
quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876
120576
119895minus 119876
119895
10038161003816100381610038161003816le 119888
1120576
10038161003816100381610038161003816119876
119902120576
1198952minus 119876
119902
1198952
10038161003816100381610038161003816le 119888
2120576 (25)
where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888
2are constants
independent of 120576Consider the system of linear algebraic equations
119906120576
ℎ= 119861119906
120576
ℎon Π
ℎ
119896
119906120576
ℎ= 120593
119898on 120578
ℎ
1198961cap 120574
119898
119906120576
ℎ(119903
119895 120579
119895)
= 119876120576
119895+ 120573
119895
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
times 119877119895(119903
119895 120579
119895 120579
119902
119895) on (119903
119895 120579
119895) isin 119905
ℎ
119896119895
119906120576
ℎ= 119878
4119906120576
ℎon 120596
ℎ119899
(26)
where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864
Abstract and Applied Analysis 5
Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2
minus984858 and 120576 = 5 times 10minus13
(2minus984858 119899) 120577
120576
ℎ119866119873119878 R
984858
119866119873119878
(2minus4 60) 1609 times 10
minus815577
(2minus5 170) 1033 times 10
minus9
(2minus5 130) 1191 times 10
minus916690
(2minus6 150) 7136 times 10
minus11
(2minus5 140) 1136 times 10
minus916259
(2minus6 170) 6991 times 10
minus11
(2minus6 100) 2169 times 10
minus1017096
(2minus7 130) 1269 times 10
minus11
Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =
2minus984858 and 120576 = 5 times 10
minus13
(2minus984858 119899) 120577
120576
ℎ119866119878 R
984858
119866119878
(2minus4 100) 1931 times 10
minus816078
(2minus5 150) 1182 times 10
minus9
(2minus5 130) 1294 times 10
minus916789
(2minus6 150) 7708 times 10
minus11
(2minus5 140) 1312 times 10
minus917967
(2minus6 170) 7304 times 10
minus11
(2minus6 100) 2389 times 10
minus1018164
(2minus7 130) 1315 times 10
minus11
Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) 120577
120576
ℎ119866119873119878 120577
120576
ℎ119866119878 Iteration
(16 70) 1139 times 10minus8
1572 times 10minus8
22
(32 170) 1033 times 10minus9
1184 times 10minus9
23
(64 170) 6990 times 10minus11
7304 times 10minus11
24
(128 200) 8628 times 10minus12
8833 times 10minus12
25
Let 119879lowast
119895= 119879
119895(119903
lowast
119895) be the sector where 119903lowast
119895= (119903
1198952+119903
1198953)2 119895 isin
119864 and let 119906120576ℎ(119903
1198952 120579
119902
119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution
values of the system (26) on 119881ℎ
119895(at the quadrature nodes)
The function
119880120576
ℎ(119903
119895 120579
119895) = 119876
119895(119903
119895 120579
119895)
+ 120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) (119906
120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
(27)
defined on 119879lowast
119895 is called an approximate solution of the
problem (1) on the closed block 1198793
119895 119895 isin 119864
Definition 5 The system (26) and (27) is called the system ofblock-grid equations
Theorem 6 There is a natural number 1198990 such that for all
119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution
Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899
0and 120590 such that for all 119899 ge 119899
0
max(119903119895120579119895)isin119879
3
119895
120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) le 120590 lt 1 (28)
The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)
Theorem 7 There exists a natural number 1198990 such that for all
119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)
where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid
max119866ℎ119899
119879
1003816100381610038161003816119906120576
ℎminus 119906
1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816le 119888
119901(ℎ
4+ 120576) on 119879
3
119895
(31)
for integer 1120572119895when 119901 ge 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895
(32)
for any 1120572119895 if 0 le 119901 lt 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895 119860
119895
(33)
for noninteger 1120572119895 when 119901 gt 1120572
119895 Everywhere 0 le 119902 le 119901 119906
is the exact solution of the problem (1) and119880120576
ℎ(119903
119895 120579
119895) is defined
by formula (27)
Proof Let
120585120576
ℎ= 119906
120576
ℎminus 119906 (34)
where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866
ℎ119899
119879
of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576
ℎsatisfies the system of difference equations
120585120576
ℎ= 119861120585
120576
ℎ+ 119903
1
ℎon Π
ℎ
119896
120585120576
ℎ= 0 on 120578
ℎ
1198961
120585120576
ℎ(119903
119895 120579
119895) = 120573
119895
119899(119895)
sum
119902=1
120585120576
ℎ(119903
1198952 120579
119902
119895) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
+ 1199032
119895ℎ (119903
119895 120579
119895) isin 119905
ℎ
119896119895
120585120576
ℎ= 119878
4120585120576
ℎ+ 119903
3
ℎon 120596
ℎ119899
(35)
6 Abstract and Applied Analysis
Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 3895 times 10minus7
3895 times 10minus7
(32 170) 4627 times 10minus8
4627 times 10minus8
(64 170) 1124 times 10minus9
3125 times 10minus9
(128 200) 2214 times 10minus10
2233 times 10minus10
Table 5 In119866119878 the minimum errors of the derivatives over the pairs
(ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 9663 times 10minus6
9663 times 10minus6
(32 170) 9653 times 10minus6
9653 times 10minus6
(64 170) 9649 times 10minus6
9649 times 10minus6
(128 200) 9648 times 10minus6
9648 times 10minus6
where 1 le 119896 le 119872 119895 isin 119864
1199031
ℎ= 119861119906 minus 119906 on cup
119872
119896=1Π
ℎ
119896 (36)
1199032
119895ℎ= 120573
119895
119899(119895)
sum
119902=1
(119906 (1199031198952 120579
119902
119895) minus 119876
119902120576
1198952) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
minus (119906 minus 119876120576
119895) on cup
119872
119896=1(cup
119895isin119864119905ℎ
119896119895)
(37)
1199033
ℎ
=
1198784119906 minus 119906 on 120596
ℎ119899
119868
1198784(119906 minus
3
sum
119896=0
119886119896Re 119911119896) minus (119906 minus
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(38)
On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows
The function 119880120576
ℎ(119903
119895 120579
119895) given by formula (27) defined
on the closed sector 119879lowast
119895 119895 isin 119864 where 119903
lowast
119895= (119903
1198952+ 119903
1198953)2
and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879
2
119895 119881
119895 119895 isin 119864 and then the
difference function 120577120576
ℎ(119903
119895 120579
119895) = 119880
120576
ℎ(119903
119895 120579
119895)minus119906(119903
119895 120579
119895) is defined
on 119879lowast
119895 119895 isin 119864 and
120577120576
ℎ(119903
lowast
119895 0) = 120577
120576
ℎ(119903
lowast
119895 120572
119895120587) = 0 119895 isin 119864 (39)
On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge
max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain
10038161003816100381610038161003816120577120576
ℎ(119903
119895 120579
119895)10038161003816100381610038161003816le 119888 (ℎ
4+ 120576) on 119879
lowast
119895 119895 isin 119864 (40)
Furthermore the function 120577120576
ℎ(119903
119895 120579
119895) continuous on 119879
lowast
119895is a
solution of the following Dirichlet problem
Δ120577120576
ℎ= 0 on 119879
lowast
119895
120577120576
ℎ= 0 on 120574
119898cap 119879
lowast
119895 119898 = 119895 minus 1 119895
120577120576
ℎ(119903
lowast
119895 120579
119895) = 119880
120576
ℎ(119903
lowast
119895 120579
119895) minus 119906 (119903
lowast
119895 120579
119895) 0 le 120579
119895le 120572
119895120587
(41)
Since 1198793
119895sub 119879
lowast
119895 on the basis of (39) and (40) from Lemma
612 in [16] inequalities (31)ndash(33) of Theorem 7 follow
5 Stress Intensity Factor
Let in the condition 120593119895isin 119862
4120582(120574
119895) the exponent 120582 be such
that
120572119895(4 + 120582) = 0 2120572
119895(4 + 120582) = 0 (42)
where sdot is the symbol of fractional partThese conditions forthe given 120572
119895can be fulfilled by decreasing 120582
On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879
lowast
119895 119895 isin 119864 as follows
119906 (119909119895 119910
119895) = (119909
119895 119910
119895) +
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911
+
119899120572119895
sum
119896=1
120591(119895)
119896119903119896120572119895
119895sin
119896120579119895
120572119895
(43)
where 119899120572119895
= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909
119895+119894119910
119895 120583(119895)
119896
and 120591(119895)
119896are some numbers and (119909
119895 119910
119895) isin 119862
4120582(119879
2
119895) is the
harmonic on 1198792
119895 By taking 120579
119895= 120572
1198951205872 from the formula
(43) it follows that the coefficient 120591(119895)1
which is called the stressintensity factor can be represented as
120591(119895)
1= lim
119903119895rarr0
1
1199031120572119895
119895
(119906 (119909119895 119910
119895) minus (119909
119895 119910
119895)
minus
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911)
(44)
Abstract and Applied Analysis 7
Table 6 The stress intensity factor 1205911205981119899
for 119899 = 70 170 200 when 120576 = 5 times 10minus13
ℎminus1
120591120576
170120591120576
1170120591120576
1200
16 1000000014856688 1000000017180415 1000000017197438
32 1000000005800844 1000000001230267 1000000001236709
64 1000000004138169 1000000000073107 1000000000079938
128 1000000004053153 1000000000003531 1000000000003267
10minus15 10minus10 10minus5 10010minus10
10minus5
100
120576
Max
erro
r
ℎ = 116 119899 = 70
(a)
10minus10
10minus5
100
10minus15 10minus10 10minus5 100
120576
Max
erro
r
ℎ = 132 119899 = 170
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 3 Dependence on 120576 for ℎminus1 = 16 32
From formula (44) it follows that 120591(119895)
1can be approxi-
mated by
120591(119895)120576
1119899
= lim119903119895rarr0
1
1199031120572119895
119895
(119880120576
ℎ(119903
119895 120579
119895)
minus(120593119895(119904
119895) + (120593
119895minus1(119904
119895) minus 120593
119895(119904
119895))
120579119895
120572119895120587))
(45)
Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula
120591(119895)120576
1119899=
1
120587int
1205901198950
0
1205931198950(119905
120572119895) 119889119905
1199052+
1
120587int
1205901198951
0
1205931198951(119905
120572119895) 119889119905
1199052
+2
119899 (119895) 1199031120572119895
1198952
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952) sin 1
120572119895
120579119902
119895
(46)
This formula is obtained for the second-order BGM in [8]
10minus15 10minus10 10minus5 100
120576
h = 164 n = 170100
10minus10
10minus20
Max
erro
r
(a)
10minus15 10minus10 10minus5 100
120576
h = 1128 n = 200100
10minus10
10minus20
Max
erro
r
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 4 Dependence on 120576 for ℎminus1 = 64 128
6 Numerical Results
Let 119866 be L-shaped and defined as follows
119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)
where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866
We consider the following problem
Δ119906 = 0 in 119866
119906 = V (119903 120579) on 120574
(48)
where
V (119903 120579) = 11990323 sin(
2
3120579) + 00051119903
163 cos(16
3120579) (49)
is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as
119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω
1 (50)
where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then
119866119873119878
= 119866 119866119878 is a ldquononsingularrdquo part of 119866
The given domain 119866 is covered by four overlappingrectangles Π
119896 119896 = 1 4 and by the block sector 119879
3
1
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2
minus984858 and 120576 = 5 times 10minus13
(2minus984858 119899) 120577
120576
ℎ119866119873119878 R
984858
119866119873119878
(2minus4 60) 1609 times 10
minus815577
(2minus5 170) 1033 times 10
minus9
(2minus5 130) 1191 times 10
minus916690
(2minus6 150) 7136 times 10
minus11
(2minus5 140) 1136 times 10
minus916259
(2minus6 170) 6991 times 10
minus11
(2minus6 100) 2169 times 10
minus1017096
(2minus7 130) 1269 times 10
minus11
Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =
2minus984858 and 120576 = 5 times 10
minus13
(2minus984858 119899) 120577
120576
ℎ119866119878 R
984858
119866119878
(2minus4 100) 1931 times 10
minus816078
(2minus5 150) 1182 times 10
minus9
(2minus5 130) 1294 times 10
minus916789
(2minus6 150) 7708 times 10
minus11
(2minus5 140) 1312 times 10
minus917967
(2minus6 170) 7304 times 10
minus11
(2minus6 100) 2389 times 10
minus1018164
(2minus7 130) 1315 times 10
minus11
Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) 120577
120576
ℎ119866119873119878 120577
120576
ℎ119866119878 Iteration
(16 70) 1139 times 10minus8
1572 times 10minus8
22
(32 170) 1033 times 10minus9
1184 times 10minus9
23
(64 170) 6990 times 10minus11
7304 times 10minus11
24
(128 200) 8628 times 10minus12
8833 times 10minus12
25
Let 119879lowast
119895= 119879
119895(119903
lowast
119895) be the sector where 119903lowast
119895= (119903
1198952+119903
1198953)2 119895 isin
119864 and let 119906120576ℎ(119903
1198952 120579
119902
119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution
values of the system (26) on 119881ℎ
119895(at the quadrature nodes)
The function
119880120576
ℎ(119903
119895 120579
119895) = 119876
119895(119903
119895 120579
119895)
+ 120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) (119906
120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952)
(27)
defined on 119879lowast
119895 is called an approximate solution of the
problem (1) on the closed block 1198793
119895 119895 isin 119864
Definition 5 The system (26) and (27) is called the system ofblock-grid equations
Theorem 6 There is a natural number 1198990 such that for all
119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution
Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899
0and 120590 such that for all 119899 ge 119899
0
max(119903119895120579119895)isin119879
3
119895
120573119895
119899(119895)
sum
119902=1
119877119895(119903
119895 120579
119895 120579
119902
119895) le 120590 lt 1 (28)
The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)
Theorem 7 There exists a natural number 1198990 such that for all
119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)
where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid
max119866ℎ119899
119879
1003816100381610038161003816119906120576
ℎminus 119906
1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816le 119888
119901(ℎ
4+ 120576) on 119879
3
119895
(31)
for integer 1120572119895when 119901 ge 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895
(32)
for any 1120572119895 if 0 le 119901 lt 1120572
119895
10038161003816100381610038161003816100381610038161003816
120597119901
120597119909119901minus119902120597119910119902(119880
120576
ℎ(119903
119895 120579
119895) minus 119906 (119903
119895 120579
119895))
10038161003816100381610038161003816100381610038161003816
le119888119901(ℎ
4+ 120576)
119903119901minus1120572119895
on 1198793
119895 119860
119895
(33)
for noninteger 1120572119895 when 119901 gt 1120572
119895 Everywhere 0 le 119902 le 119901 119906
is the exact solution of the problem (1) and119880120576
ℎ(119903
119895 120579
119895) is defined
by formula (27)
Proof Let
120585120576
ℎ= 119906
120576
ℎminus 119906 (34)
where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866
ℎ119899
119879
of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576
ℎsatisfies the system of difference equations
120585120576
ℎ= 119861120585
120576
ℎ+ 119903
1
ℎon Π
ℎ
119896
120585120576
ℎ= 0 on 120578
ℎ
1198961
120585120576
ℎ(119903
119895 120579
119895) = 120573
119895
119899(119895)
sum
119902=1
120585120576
ℎ(119903
1198952 120579
119902
119895) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
+ 1199032
119895ℎ (119903
119895 120579
119895) isin 119905
ℎ
119896119895
120585120576
ℎ= 119878
4120585120576
ℎ+ 119903
3
ℎon 120596
ℎ119899
(35)
6 Abstract and Applied Analysis
Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 3895 times 10minus7
3895 times 10minus7
(32 170) 4627 times 10minus8
4627 times 10minus8
(64 170) 1124 times 10minus9
3125 times 10minus9
(128 200) 2214 times 10minus10
2233 times 10minus10
Table 5 In119866119878 the minimum errors of the derivatives over the pairs
(ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 9663 times 10minus6
9663 times 10minus6
(32 170) 9653 times 10minus6
9653 times 10minus6
(64 170) 9649 times 10minus6
9649 times 10minus6
(128 200) 9648 times 10minus6
9648 times 10minus6
where 1 le 119896 le 119872 119895 isin 119864
1199031
ℎ= 119861119906 minus 119906 on cup
119872
119896=1Π
ℎ
119896 (36)
1199032
119895ℎ= 120573
119895
119899(119895)
sum
119902=1
(119906 (1199031198952 120579
119902
119895) minus 119876
119902120576
1198952) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
minus (119906 minus 119876120576
119895) on cup
119872
119896=1(cup
119895isin119864119905ℎ
119896119895)
(37)
1199033
ℎ
=
1198784119906 minus 119906 on 120596
ℎ119899
119868
1198784(119906 minus
3
sum
119896=0
119886119896Re 119911119896) minus (119906 minus
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(38)
On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows
The function 119880120576
ℎ(119903
119895 120579
119895) given by formula (27) defined
on the closed sector 119879lowast
119895 119895 isin 119864 where 119903
lowast
119895= (119903
1198952+ 119903
1198953)2
and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879
2
119895 119881
119895 119895 isin 119864 and then the
difference function 120577120576
ℎ(119903
119895 120579
119895) = 119880
120576
ℎ(119903
119895 120579
119895)minus119906(119903
119895 120579
119895) is defined
on 119879lowast
119895 119895 isin 119864 and
120577120576
ℎ(119903
lowast
119895 0) = 120577
120576
ℎ(119903
lowast
119895 120572
119895120587) = 0 119895 isin 119864 (39)
On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge
max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain
10038161003816100381610038161003816120577120576
ℎ(119903
119895 120579
119895)10038161003816100381610038161003816le 119888 (ℎ
4+ 120576) on 119879
lowast
119895 119895 isin 119864 (40)
Furthermore the function 120577120576
ℎ(119903
119895 120579
119895) continuous on 119879
lowast
119895is a
solution of the following Dirichlet problem
Δ120577120576
ℎ= 0 on 119879
lowast
119895
120577120576
ℎ= 0 on 120574
119898cap 119879
lowast
119895 119898 = 119895 minus 1 119895
120577120576
ℎ(119903
lowast
119895 120579
119895) = 119880
120576
ℎ(119903
lowast
119895 120579
119895) minus 119906 (119903
lowast
119895 120579
119895) 0 le 120579
119895le 120572
119895120587
(41)
Since 1198793
119895sub 119879
lowast
119895 on the basis of (39) and (40) from Lemma
612 in [16] inequalities (31)ndash(33) of Theorem 7 follow
5 Stress Intensity Factor
Let in the condition 120593119895isin 119862
4120582(120574
119895) the exponent 120582 be such
that
120572119895(4 + 120582) = 0 2120572
119895(4 + 120582) = 0 (42)
where sdot is the symbol of fractional partThese conditions forthe given 120572
119895can be fulfilled by decreasing 120582
On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879
lowast
119895 119895 isin 119864 as follows
119906 (119909119895 119910
119895) = (119909
119895 119910
119895) +
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911
+
119899120572119895
sum
119896=1
120591(119895)
119896119903119896120572119895
119895sin
119896120579119895
120572119895
(43)
where 119899120572119895
= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909
119895+119894119910
119895 120583(119895)
119896
and 120591(119895)
119896are some numbers and (119909
119895 119910
119895) isin 119862
4120582(119879
2
119895) is the
harmonic on 1198792
119895 By taking 120579
119895= 120572
1198951205872 from the formula
(43) it follows that the coefficient 120591(119895)1
which is called the stressintensity factor can be represented as
120591(119895)
1= lim
119903119895rarr0
1
1199031120572119895
119895
(119906 (119909119895 119910
119895) minus (119909
119895 119910
119895)
minus
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911)
(44)
Abstract and Applied Analysis 7
Table 6 The stress intensity factor 1205911205981119899
for 119899 = 70 170 200 when 120576 = 5 times 10minus13
ℎminus1
120591120576
170120591120576
1170120591120576
1200
16 1000000014856688 1000000017180415 1000000017197438
32 1000000005800844 1000000001230267 1000000001236709
64 1000000004138169 1000000000073107 1000000000079938
128 1000000004053153 1000000000003531 1000000000003267
10minus15 10minus10 10minus5 10010minus10
10minus5
100
120576
Max
erro
r
ℎ = 116 119899 = 70
(a)
10minus10
10minus5
100
10minus15 10minus10 10minus5 100
120576
Max
erro
r
ℎ = 132 119899 = 170
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 3 Dependence on 120576 for ℎminus1 = 16 32
From formula (44) it follows that 120591(119895)
1can be approxi-
mated by
120591(119895)120576
1119899
= lim119903119895rarr0
1
1199031120572119895
119895
(119880120576
ℎ(119903
119895 120579
119895)
minus(120593119895(119904
119895) + (120593
119895minus1(119904
119895) minus 120593
119895(119904
119895))
120579119895
120572119895120587))
(45)
Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula
120591(119895)120576
1119899=
1
120587int
1205901198950
0
1205931198950(119905
120572119895) 119889119905
1199052+
1
120587int
1205901198951
0
1205931198951(119905
120572119895) 119889119905
1199052
+2
119899 (119895) 1199031120572119895
1198952
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952) sin 1
120572119895
120579119902
119895
(46)
This formula is obtained for the second-order BGM in [8]
10minus15 10minus10 10minus5 100
120576
h = 164 n = 170100
10minus10
10minus20
Max
erro
r
(a)
10minus15 10minus10 10minus5 100
120576
h = 1128 n = 200100
10minus10
10minus20
Max
erro
r
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 4 Dependence on 120576 for ℎminus1 = 64 128
6 Numerical Results
Let 119866 be L-shaped and defined as follows
119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)
where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866
We consider the following problem
Δ119906 = 0 in 119866
119906 = V (119903 120579) on 120574
(48)
where
V (119903 120579) = 11990323 sin(
2
3120579) + 00051119903
163 cos(16
3120579) (49)
is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as
119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω
1 (50)
where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then
119866119873119878
= 119866 119866119878 is a ldquononsingularrdquo part of 119866
The given domain 119866 is covered by four overlappingrectangles Π
119896 119896 = 1 4 and by the block sector 119879
3
1
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878cap 119903ge02
11990313
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 3895 times 10minus7
3895 times 10minus7
(32 170) 4627 times 10minus8
4627 times 10minus8
(64 170) 1124 times 10minus9
3125 times 10minus9
(128 200) 2214 times 10minus10
2233 times 10minus10
Table 5 In119866119878 the minimum errors of the derivatives over the pairs
(ℎminus1 119899) in maximum norm when 120576 = 5 times 10
minus13
(ℎminus1 119899) Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119909minus
120597119906
120597119909
10038171003817100381710038171003817100381710038171003817Max
119866119878119903
13
10038171003817100381710038171003817100381710038171003817
120597119880120576
ℎ
120597119910minus
120597119906
120597119910
10038171003817100381710038171003817100381710038171003817
(16 70) 9663 times 10minus6
9663 times 10minus6
(32 170) 9653 times 10minus6
9653 times 10minus6
(64 170) 9649 times 10minus6
9649 times 10minus6
(128 200) 9648 times 10minus6
9648 times 10minus6
where 1 le 119896 le 119872 119895 isin 119864
1199031
ℎ= 119861119906 minus 119906 on cup
119872
119896=1Π
ℎ
119896 (36)
1199032
119895ℎ= 120573
119895
119899(119895)
sum
119902=1
(119906 (1199031198952 120579
119902
119895) minus 119876
119902120576
1198952) 119877
119895(119903
119895 120579
119895 120579
119902
119895)
minus (119906 minus 119876120576
119895) on cup
119872
119896=1(cup
119895isin119864119905ℎ
119896119895)
(37)
1199033
ℎ
=
1198784119906 minus 119906 on 120596
ℎ119899
119868
1198784(119906 minus
3
sum
119896=0
119886119896Re 119911119896) minus (119906 minus
3
sum
119896=0
119886119896Re 119911119896)
119875
119875 isin 120596ℎ119899
119863
(38)
On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows
The function 119880120576
ℎ(119903
119895 120579
119895) given by formula (27) defined
on the closed sector 119879lowast
119895 119895 isin 119864 where 119903
lowast
119895= (119903
1198952+ 119903
1198953)2
and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879
2
119895 119881
119895 119895 isin 119864 and then the
difference function 120577120576
ℎ(119903
119895 120579
119895) = 119880
120576
ℎ(119903
119895 120579
119895)minus119906(119903
119895 120579
119895) is defined
on 119879lowast
119895 119895 isin 119864 and
120577120576
ℎ(119903
lowast
119895 0) = 120577
120576
ℎ(119903
lowast
119895 120572
119895120587) = 0 119895 isin 119864 (39)
On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge
max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain
10038161003816100381610038161003816120577120576
ℎ(119903
119895 120579
119895)10038161003816100381610038161003816le 119888 (ℎ
4+ 120576) on 119879
lowast
119895 119895 isin 119864 (40)
Furthermore the function 120577120576
ℎ(119903
119895 120579
119895) continuous on 119879
lowast
119895is a
solution of the following Dirichlet problem
Δ120577120576
ℎ= 0 on 119879
lowast
119895
120577120576
ℎ= 0 on 120574
119898cap 119879
lowast
119895 119898 = 119895 minus 1 119895
120577120576
ℎ(119903
lowast
119895 120579
119895) = 119880
120576
ℎ(119903
lowast
119895 120579
119895) minus 119906 (119903
lowast
119895 120579
119895) 0 le 120579
119895le 120572
119895120587
(41)
Since 1198793
119895sub 119879
lowast
119895 on the basis of (39) and (40) from Lemma
612 in [16] inequalities (31)ndash(33) of Theorem 7 follow
5 Stress Intensity Factor
Let in the condition 120593119895isin 119862
4120582(120574
119895) the exponent 120582 be such
that
120572119895(4 + 120582) = 0 2120572
119895(4 + 120582) = 0 (42)
where sdot is the symbol of fractional partThese conditions forthe given 120572
119895can be fulfilled by decreasing 120582
On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879
lowast
119895 119895 isin 119864 as follows
119906 (119909119895 119910
119895) = (119909
119895 119910
119895) +
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911
+
119899120572119895
sum
119896=1
120591(119895)
119896119903119896120572119895
119895sin
119896120579119895
120572119895
(43)
where 119899120572119895
= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909
119895+119894119910
119895 120583(119895)
119896
and 120591(119895)
119896are some numbers and (119909
119895 119910
119895) isin 119862
4120582(119879
2
119895) is the
harmonic on 1198792
119895 By taking 120579
119895= 120572
1198951205872 from the formula
(43) it follows that the coefficient 120591(119895)1
which is called the stressintensity factor can be represented as
120591(119895)
1= lim
119903119895rarr0
1
1199031120572119895
119895
(119906 (119909119895 119910
119895) minus (119909
119895 119910
119895)
minus
4
sum
119896=0
120583(119895)
119896Im 119911
119896 ln 119911)
(44)
Abstract and Applied Analysis 7
Table 6 The stress intensity factor 1205911205981119899
for 119899 = 70 170 200 when 120576 = 5 times 10minus13
ℎminus1
120591120576
170120591120576
1170120591120576
1200
16 1000000014856688 1000000017180415 1000000017197438
32 1000000005800844 1000000001230267 1000000001236709
64 1000000004138169 1000000000073107 1000000000079938
128 1000000004053153 1000000000003531 1000000000003267
10minus15 10minus10 10minus5 10010minus10
10minus5
100
120576
Max
erro
r
ℎ = 116 119899 = 70
(a)
10minus10
10minus5
100
10minus15 10minus10 10minus5 100
120576
Max
erro
r
ℎ = 132 119899 = 170
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 3 Dependence on 120576 for ℎminus1 = 16 32
From formula (44) it follows that 120591(119895)
1can be approxi-
mated by
120591(119895)120576
1119899
= lim119903119895rarr0
1
1199031120572119895
119895
(119880120576
ℎ(119903
119895 120579
119895)
minus(120593119895(119904
119895) + (120593
119895minus1(119904
119895) minus 120593
119895(119904
119895))
120579119895
120572119895120587))
(45)
Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula
120591(119895)120576
1119899=
1
120587int
1205901198950
0
1205931198950(119905
120572119895) 119889119905
1199052+
1
120587int
1205901198951
0
1205931198951(119905
120572119895) 119889119905
1199052
+2
119899 (119895) 1199031120572119895
1198952
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952) sin 1
120572119895
120579119902
119895
(46)
This formula is obtained for the second-order BGM in [8]
10minus15 10minus10 10minus5 100
120576
h = 164 n = 170100
10minus10
10minus20
Max
erro
r
(a)
10minus15 10minus10 10minus5 100
120576
h = 1128 n = 200100
10minus10
10minus20
Max
erro
r
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 4 Dependence on 120576 for ℎminus1 = 64 128
6 Numerical Results
Let 119866 be L-shaped and defined as follows
119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)
where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866
We consider the following problem
Δ119906 = 0 in 119866
119906 = V (119903 120579) on 120574
(48)
where
V (119903 120579) = 11990323 sin(
2
3120579) + 00051119903
163 cos(16
3120579) (49)
is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as
119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω
1 (50)
where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then
119866119873119878
= 119866 119866119878 is a ldquononsingularrdquo part of 119866
The given domain 119866 is covered by four overlappingrectangles Π
119896 119896 = 1 4 and by the block sector 119879
3
1
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
Table 6 The stress intensity factor 1205911205981119899
for 119899 = 70 170 200 when 120576 = 5 times 10minus13
ℎminus1
120591120576
170120591120576
1170120591120576
1200
16 1000000014856688 1000000017180415 1000000017197438
32 1000000005800844 1000000001230267 1000000001236709
64 1000000004138169 1000000000073107 1000000000079938
128 1000000004053153 1000000000003531 1000000000003267
10minus15 10minus10 10minus5 10010minus10
10minus5
100
120576
Max
erro
r
ℎ = 116 119899 = 70
(a)
10minus10
10minus5
100
10minus15 10minus10 10minus5 100
120576
Max
erro
r
ℎ = 132 119899 = 170
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 3 Dependence on 120576 for ℎminus1 = 16 32
From formula (44) it follows that 120591(119895)
1can be approxi-
mated by
120591(119895)120576
1119899
= lim119903119895rarr0
1
1199031120572119895
119895
(119880120576
ℎ(119903
119895 120579
119895)
minus(120593119895(119904
119895) + (120593
119895minus1(119904
119895) minus 120593
119895(119904
119895))
120579119895
120572119895120587))
(45)
Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula
120591(119895)120576
1119899=
1
120587int
1205901198950
0
1205931198950(119905
120572119895) 119889119905
1199052+
1
120587int
1205901198951
0
1205931198951(119905
120572119895) 119889119905
1199052
+2
119899 (119895) 1199031120572119895
1198952
119899(119895)
sum
119902=1
(119906120576
ℎ(119903
1198952 120579
119902
119895) minus 119876
119902120576
1198952) sin 1
120572119895
120579119902
119895
(46)
This formula is obtained for the second-order BGM in [8]
10minus15 10minus10 10minus5 100
120576
h = 164 n = 170100
10minus10
10minus20
Max
erro
r
(a)
10minus15 10minus10 10minus5 100
120576
h = 1128 n = 200100
10minus10
10minus20
Max
erro
r
||120577h||(G119873119878)||120577h||(G119878)
(b)
Figure 4 Dependence on 120576 for ℎminus1 = 64 128
6 Numerical Results
Let 119866 be L-shaped and defined as follows
119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)
where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866
We consider the following problem
Δ119906 = 0 in 119866
119906 = V (119903 120579) on 120574
(48)
where
V (119903 120579) = 11990323 sin(
2
3120579) + 00051119903
163 cos(16
3120579) (49)
is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as
119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω
1 (50)
where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then
119866119873119878
= 119866 119866119878 is a ldquononsingularrdquo part of 119866
The given domain 119866 is covered by four overlappingrectangles Π
119896 119896 = 1 4 and by the block sector 119879
3
1
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
0 50 100 150 200 250n
10minus8
10minus7
10minus6
Max
erro
r
h = 116
(a)
0 50 100 150 200 250n
Max
erro
r
10minus10
10minus8
10minus6h = 132
(b)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 164
||120577h||(G119873119878)||120577h||(G119878)
(c)
0 50 100 150 200 250n
10minus15
10minus10
10minus5
Max
erro
r
h = 1128
||120577h||(G119873119878)||120577h||(G119878)
(d)
Figure 5 Maximum error depending on the number of quadrature nodes 119899
0
02
04
06
08
U120576 h
0 0
0505
minus05 minus05x-axis
y-axis
(a)
0
02
04
06
08
u
0 0
0505
minus05 minus05x-axis
y-axis
(b)
Figure 6 The approximate solution 119880120576
ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10
minus13
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
0
2
4
6
8times10minus11
0 0
0505
minus05 minus05
|120577h| (G119878)
x-axisy-axis
Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13
minus2
minus15
minus1
minus05
0
120597U120576 h120597x
0 0
0505
minus05 minus05 x-axisy-axis
Figure 8 120597119880120576
ℎ120597
119909in the ldquosingularrdquo part
(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905
1= 119886119887119888119889119890 The radius 119903
12of sector 1198792
1is taken as 093
According to (49) the function 119876(119903 120579) in (4) is
119876 (119903 120579) =00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
+00051
120587int
1
0
1199101199058119889119905
(119905 minus 119909)2+ 1199102
(51)
where 119909 = 11990323 cos(21205793) and 119910 = 119903
23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876
120576(119903
12 120579
119902) and 119876
120576(119903 120579) on the grids 119905
ℎ
1
with an accuracy of 120576using the quadrature formulae proposedin [10]
On the basis of (46) and (51) for the stress intensity factorwe have
120591120576
1119899=
00102
7120587+
2
119899(093)23
119899
sum
119902=1
(119906120576
ℎ(093 120579
119902
119895) minus 119876
119902120576
1198952) sin 2
3120579119902
119895
(52)
Taking the zero approximation 119906120576(0)
ℎ= 0 the results of
realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables
0
05
1
15
2
120597U120576 h120597y
0 0
0505
minus05 minus05 x-axisy-axis
Figure 9 120597119880120576
ℎ120597
119910in the ldquosingularrdquo part
40 60 80 100 120 140 160 180 200 2200
5
10
15
20
25
30
n
n = 80n = 90n = 100n = 130n = 150
n = 170n = 180n = 200n = 220
Rϱ G119878
Figure 10 R984858
119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
0
5
10
15
20
25
40 60 80 100 120 140 160 180 200 220n
n = 80
n = 100
n = 120
n = 130
n = 140
n = 170
n = 190
n = 200
n = 220
Rϱ G119873119878
Figure 11R984858
119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899
values of ℎminus1 = 64
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM
R984858
119866119873119878 =
max119866119873119878
1003816100381610038161003816119906120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119873119878
100381610038161003816100381610038161199061205762minus(984858+1) minus 119906
10038161003816100381610038161003816
(53)
in the ldquononsingularrdquo and the order of convergence
R984858
119866119878 =
max1198661198781003816100381610038161003816119880
120576
2minus984858 minus 119906
1003816100381610038161003816
max119866119878
10038161003816100381610038161003816119880120576
2minus(984858+1) minus 119906
10038161003816100381610038161003816
(54)
in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times
10minus13 where 984858 is a positive integer In Table 3 the minimal
values over the pairs (ℎminus1 119899) of the errors in maximum
norm of the approximate solution when 120576 = 5 times 10minus13
are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910
are approximated by fourth-order central difference formulaon 119866
119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10
minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also
Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10
minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858
119866119878 and R
984858
119866119873119878
when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1
= 32 and it is selected as large as possible (119899 gt 100) forℎminus1
= 64
7 Conclusions
In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862
4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order
119874(ℎ4+ 120576) where 120576 is the error of the approximation of
the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ
4+120576)119903
1120572119895minus119901
119895) order is obtainedThemethod is illustrated
in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given
From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895
119895(ℎ
4+ 120576) which gives an additional
accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply
connected polygons
References
[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998
[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004
[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992
[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994
[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004
[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005
[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007
[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010
[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012
[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993
[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965
[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001
[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003
[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002
[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980
[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994
[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of